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T H E R I A 

MOTVS CORPORVM 

COELESTIYM 



IN 



SECTIONIBVS CONICIS SOLEM AMBIENTIVM 



A V C T O R E 



CAROLO FRIDERICO GAVSS. 



THEORY 



MOTION OF THE HEAVENLY BODIES MOVING ABOUT 
THE SUN IN CONIC SECTIONS: 



A TBAN8LATION OP 



GAUSS'S "THEORIA MOTUS." 



WITH AN APPENDIX. 



CHARLES HENRY DAVIS, 

COMMANDER UNITED STATES NAVY, SUPEKINTBNDENT OP THE AMERICAN EPHEMERIS AND NAUTICAL ALMANAC. 



BOSTON: 

LITTLE, BROWN AND COMPANY. 

185 7. 



Published under the Authority of the Navy Department by the Nautical Almanac and 
Smithsonian Institution. 



TRANSLATOE'S PREFACE. 



In 1852, a pamphlet, entitled The Computation of an Orbit from Three Complete 
Observations, was published, under the authority of the Navy Department, for the use 
of the American Ephemeris and Nautical Almanac, the object of which was to excerpt 
from various parts of Gauss's Theoria Motus, and to arrange in proper order the numer- 
ous details which combine to form this complicated problem. To these were added an 
Appendix containing the results of Professor Eistcke's investigations, Ueber den Ausnah- 
mefall einer doppelten Bahnbestimmung aus denselben drei geocentrischen Oertern (Ab- 
handlungen der Akademie der Wissenschaften zu Berlin, 1848), and also Professor Peirce's 
Graphic Delineations of the Curves showing geometrically the roots of Gauss's Equa- 
tion IV. Article 141. 

After this pamphlet was completed, the opinion was expressed by scientific friends 
that a complete translation of the Theoria Motus should be undertaken, not only to meet 
the wants of the American Ephemeris, but those also of Astronomers generally, to whom 
this work (now become very rare and costly) is a standard and permanent authority. 
This undertaking has been particularly encouraged by the Smithsonian Institution, 
which has signified its high estimate of the importance of the work, by contributing to 
its publication. And by the authority of Hon. J. C. Dobbin, Secretary of the Navy, this 
Translation is printed by the joint contributions of the Nautical Almanac and the Smith- 
sonian Institution. 

The notation of Gauss has been strictly adhered to throughout, and the translation 
has been made as nearly literal as possible. No pains have been spared to secure typo- 
graphical accuracy. All the errata that have been noticed in Zach's Monatliche Corre- 
spondenz, the Berliner Astronomisches Jahrbuch, and the Astronomische Nachrichten, have 

(V) 



VI TRANSLATOR'S PREFACE. 

been corrected, and in addition to these a considerable number, a list of which will be 
found in Gould's Astronomical Journal, that were discovered by Professor Chadve:n'et 
of the United States Naval Academy, who has examined the formulas of the body of 
the work with great care, not only by comparison with the original, but by independent 
verification. The proof-sheets have also been carefully read by Professor Phillips, of 
Chapel Hill, North Carolina, and by Mr, Runkle and Professor Winlock of the Nautical 
Almanac office. 

The Appendix contains the results of the investigations of Professor Encke and 
Professor Peirce, from the Appendix of the pamphlet above referred to, and other mat- 
ters which, it is hoped, will be found interesting and useful to the practical computer, 
among which are several valuable tables : A Table for the Motion in a Parabola from 
LeVerrier's Annates de U Observatoire Imperial de Paris, Bessel's and Posselt's 
Tables for Ellipses and Hyperbolas closely resembling the Parabola, and a convenient 
Table by Professor Hubbard for facilitating the use of Gauss's formulas for Ellipses and 
Hyperbolas of which the eccentricities are nearly equal to unity. And in the form of 
notes on their appropriate articles, useful formulas by Bessel, Nicolai, Encke, Gauss, 
and Peirce, and a summary of the formulas for computing the orbit of a Comet, 
with the accompanying Table, from Olbers's Abhandlung ueber die leichteste und be- 
quemste Methode die Balm eines Cometen zu berechnen. "Weimar, 1847. 



CONTENTS 



Page 

Pkeface ix 



FIKST BOOK. 

GENERAL RELATIONS BETWEEN THE QUANTITIES BY WHICH THE MOTIONS 
OF HEAVENLY BODIES ABOUT THE SUN ARE DEFINED. 

First Section. — Relations pertaining simply to position in the Orbit 1 

Second Section. — Relations pertaining simply to Position in Space . . . . • 54 

Third Section. — Relations between Several Places in Orbit ....... 100 

Fourth Section. — Relations between Several Places in Space 153 



SECOND BOOK. 

INVESTIGATION OF THE ORBITS OF HEAVENLY BODIES FROM GEOCENTRIC 
OBSERVATIONS. 

First Section. — Determination of an Orbit from Three Complete Observations . . .161 
Second Section. — Determination of an Orbit from Fom- Observations, of which Two only 

are Complete 234 

Third Section. — Determination of an Orbit satisfying as nearly aS possible any number of 

Observations whatever 2^9 

Fourth Section. — On the Determination of Orbits, taking into accomit the Perturbations . 274 

Appendix 279 

Tables 329 

(vii) 



CAMBRIDGE : 
PEINTED BT ALLEN AND FARNHi 



PREFACE. 



After the laws of planetary motion were discovered, the genius of Kepler 
was not without resources for deriving from observations the elements of mo- 
tion of individual 'planets. Tycho Brake, by whom practical astronomy had 
been carried to a degree of perfection before imknown, had observed all the 
planets through a long series of years with the greatest care, and with so 
much perseverance, that there remained to Kepler, the most worthy inheritor 
of such a repository, the trouble only of selecting what might seem suited 
to any special purpose. The mean motions of the planets already deter- 
mined with great precision by means of very ancient observations diminished 
not a little this labor. 

Astronomers who, subsequently to Kepler, endeavored to determine still 
more accurately the orbits of the planets with the aid of more recent or 
better observations, enjoyed the same or even greater facilities. For the 
problem was no longer to deduce elements wholly unknown, but only 
slightly to correct those already known, and to define them within narrower 
limits. 

The principle of universal gravitation discovered by the illustrious Newton 



X PREFACE. 

opened a field entirely new, and showed that all the heavenly bodJes, at 
least those the motions of which are regulated by the attraction of the sun, 
must necessarily, conform to the same laws, with a slight modification only, 
by which Ej:pler had found the five planets to be governed. Kepler, rely- 
ing upon the evidence of observations, had announced that the orbit of every 
planet is an ellipse, in which the areas are described uniformly about the 
sun occupying one focus of the ellipse, and in such a manner that in differ- 
ent ellipses the times of revolution are in the sesquialteral ratio of the semi- 
axes-major. On the other hand, Newton, starting from the principle of 
universal gravitation, demonstrated d, priori that all bodies controlled by the 
attractive force of the sun must move in conic sections, of which the planets 
present one form to us, namely, ellipses, while the remaining forms, parabo- 
las and hyperbolas, must be regarded as being equally possible, provided 
there may be bodies encountering the force of the sun with the requisite 
velocity; that the sun must always occupy one focus of the conic section; 
that the areas which the same body describes in different times about the 
sun are proportional to those times; and finally, that the areas described 
about the sun by different bodies, in equal times, are in the subduplicate 
ratio of the semiparameters of the orbits: the latter of these laws, identical 
in elliptic motion with the last law of Kepler, extends to the parabolic and 
hyperbolic motion, to which Kepler's law cannot be applied, because the rev- 
olutions are wanting. The clue was now discovered by following which it 
became possible to enter the hitherto inaccessible labyrinth of the motions of 
the comets. And this was so successful that the single hjqsothesis, that their 
orbits were parabolas, sufficed to explain the motions of all the comets which 
had been accurately observed. Thus the system of universal gravitation had 



PREFACE. Xi 

paved' the way to new and most brilliant triumphs in analysis j and the 
comets, up to that time wholly unmanageable, or soon breaking from the 
restraints to which they seemed to be subjected, having now submitted to 
control, and being transformed from enemies to guests, moved on in the 
paths marked out by the calculus, scrupulously conforming to the same eter- 
nal laws that govern the planets. 

In determining the parabolic orbits of comets from observation, difficul- 
ties arose far greater than in determining the elliptic orbits of planets, and 
principally from this source, that comets, seen for a brief interval, did not 
afford a choice of observations particularly suited to a given object : but the 
geometer was compelled to employ those which happened to be furnished 
him, so that it became necessary to make use of special methods seldom 
applied in planetary calculations. The great Newton himself, the first geome- 
ter of his age, did not disguise the difficulty of the problem: as might have 
been expected, he came out of this contest also the victor. Since the time 
of Newton, many geometers have labored zealously on the same problem, 
with various success, of course, but still in such a manner as to leave but 
little to be desired at the present time. 

The truth, however, is not to be overlooked that in this problem the 
difficulty is very fortunately lessened by the knowledge of one element of 
the conic section, since the major-axis is put equal to infinity by the very 
assumption of the parabolic orbit. For, all parabolas, if position is neg- 
lected, differ among themselves only by the greater or less distance of the 
vertex from the focus; while conic sections, generally considered, admit of 
infinitely greater variety. There existed, in point of fact, no sufficient reason 
why it should be taken for granted that the paths of comets are exactly 



xii PREFACE. 

parabolic: on the contrary, it must be regarded as in the highest degree 
improbable that nature should ever have favored such an hypothesis. Since, 
nevertheless, it was known, that the phenomena of a heavenly body moving 
in an ellipse or hyperbola, the major-axis of which is very great relatively to 
the parameter, differs very little near the perihelion from the motion in a 
parabola of which the vertex is at the same distance from the focus; and 
that this difference becomes the more inconsiderable the greater the ratio of 
the axis to the parameter : and since, moreover, experience had shown that 
between the observed motion and the motion computed in the parabolic 
orbit, there remained differences scarcely ever greater than those which might 
safely be attributed to errors of observation (errors quite considerable in 
most cases) : astronomers have thought proper to retain the parabola, and 
very properly, because there are no means whatever of ascertaining satis- 
factorily what, if any, are the differences from a parabola. We must except 
the celebrated comet of Halley, which, describing a very elongated ellipse and 
frequently observed at its return to the perihelion, revealed to us its periodic 
time ; but then the major-axis being thus known, the computation of the re- 
maining elements is to be considered as hardly more difficult than the determi- 
nation of the parabolic orbit. And we must not omit to mention that astrono- 
mers, in the case of some other comets observed for a somewhat longer time, 
have attempted to determine the deviation from a parabola. However, all 
the methods either proposed or used for this object, rest upon the assumption 
that the variation from a parabola is inconsiderable, and hence in the trials 
referred to, the parabola itself, previously computed, furnished an approximate 
idea of the several elements (except the major-axis, or the time of revolu- 
tion depending on it), to be corrected by only slight changes. Besides, it 



PREFACE. XIU 

must be acknowledged, that the whole of these trials hardly served in any 
case to settle any thing with certainty, if, perhaps, the comet of the year 
1770 is excepted. 

As soon as it was ascertained that the motion of the new planet, discov- 
ered in 1781, could not be reconciled with the parabolic hypothesis, astrono- 
mers undertook to adapt a circular orbit to it, which is a matter of simple 
and very easy calculation. By a happy accident the orbit of this planet had 
but a small eccentricity, in consequence of which the elements resulting from 
the circular hypothesis sufficed at least for an approximation on which could 
be based the determination of the elliptic elements. There was a concur- 
rence of several other very favorable circumstances. For, the slow motion of 
the planet, and the very small inclination of the orbit to the plane of the 
ecliptic, not only rendered the calculations much more simple, and allowed 
the use of special methods not suited to other cases; but they removed the 
apprehension, lest the planet, lost in the rays of the sun, should subsequently 
elude the search of observers, (an apprehension which some astronomers might 
have felt, especially if its light had been less brilliant) ; so that the more 
accurate determination of the orbit might be safely deferred, until a selection 
could be made from observations more frequent and more remote, such as 
seemed best fitted for the end in view. 

Thus, in every case in which it was necessary to deduce the orbits of 
heavenly bodies from observations, there existed advantages not to be de- 
spised, suggesting, or at any rate permitting, the application of special 
methods; of which advantages the chief one was, that by means of hypo- 
thetical assumptions an approximate knowledge of some elements could be 



XIV PREFACE. 

obtained before the computation of the elliptic elements was commenced. 
Notwithstanding this, it seems somewhat strange that the general problem, — 

To determine the orhit of a heavenly hodt/, ivithout any hypothetical assumption, 
from observations not embracing a great period ,of time, and not alloimig a selection 
tvith a view to the application of special methods, was almost wholly neglected up 
to the beginning of the present century j or, at least, not treated by any one 
in a manner worthy of its importance; since it assuredly commended itself 
to mathematicians by its difficulty and elegance, even if its great utility in 
practice were not apparent. An opinion had universally prevailed that a 
complete determination from observations embracing a short interval of time 
was impossible, — an ill-founded opinion, — for it is now clearly shown that 
the orbit of a heavenly body may be determined quite nearly from good 
observations embracing only a few days; and this without any hypothetical 
assumption. 

Some ideas occurred to me in the month of September of the year 1801, 
engaged at the time on a very different subject, which seemed to point to 
the solution of the great problem of which I have spoken. Under such cir- 
cumstances we not unfrequently, for fear of being too much led away by 
an attractive investigation, suffer the associations of ideas, which, more atten- 
tively considered, might have proved most fruitful in results, to be lost from 
neglect. And the same fate might have befallen these conceptions, had they 
not happily occurred at the most propitious moment for their preservation 
and encouragement that could have been selected. For just about this time 
the report of the new planet, discovered on the first day of January of that 
year with the telescope at Palermo, was the subject of universal conversation; 



PREFACE. XV 

and soon afterwards the observations made by that distinguished astronomer 
PiAZZi from the above date to the eleventh of February were pubHshed. No- 
where in the annals of astronomy do we meet with so great an opportunity, 
and a greater one could hardly be imagined, for showing most strikingly, the 
value of this problem, than in this crisis and urgent necessity, when all hope 
of discovering in the heavens this planetary atom, among innumerable small 
stars after the lapse of nearly a year, rested solely upon a sufficiently ap- 
proximate knowledge of its orbit to be based upon these very few observa- 
tions. Could I ever have found a more seasonable opportunity to test the 
practical value of my conceptions, than now in employing them for the de- 
termination of the orbit of the planet Ceres, which during these forty-one 
days had described a geocentric arc of only three degrees, and after the 
lapse of a year must be looked for in a region of the heavens very remote 
from that in which it was last seen ? This first application of the method 
was made in the month of October, 1801, and the first clear night, when 
the planet was sought for* as directed by the numbers deduced from it, re- 
stored the fugitive to observation. Three other new planets, subsequently 
discovered, furnished new opportunities for examining and verifying the efii- 
ciency and generality of the method. 

Several astronomers wished me to publish the methods employed in these 
calculations immediately after the second discovery of Ceres ; but many 
things — other occupations, the desire of treating the subject more fully at 
some subsequent period, and, especially, the hope that a further prosecution 
of this investigation would raise various parts of the solution to a greater 



By de Zach, December 7, 1801. 
2 



Xvi PREFACE. 

degree of generality, simplicity, and elegance, — prevented my complying at 
the time with these friendly solicitations. I was not disappointed in this ex- 
pectation, and have no cause to regret the delay. For, the methods first 
employed have undergone so many and such great changes, that scarcely 
any trace of resemblance remains between the method in which the orbit of 
Ceres was first computed, and the form given in this work. Although it 
would be foreign to my purpose, to narrate in detail all the steps by 
which these investigations have been gradually perfected, still, in several 
instances, particularly when the problem was one of more importance than 
usual, I have thought that the earlier methods ought not to be wholly sup- 
pressed. But in this work, besides the solutions of the principal problems, 
I have given many things which, during the long time I have been en- 
gaged upon the motions of the heavenly bodies in conic sections, struck 
me as worthy of attention, either on account of their analytical elegance, 
or more especially on account of their practical utility. But in every case 
I have devoted greater care both to the subjects and methods which are 
peculiar to myself, touching lightly and so far only as the connection seemed 
to require, on those previously known. 

The whole work is divided into two parts. In the First Book are de- 
veloped the relations between the quantities on which the motion of the 
heavenly bodies about the sun, according to the laws of Keplek, depends; 
the two first sections comprise those relations in which one place only is 
considered, and the third and fourth sections those in which the relations 
between several places are considered. The two latter contain an explanation 
of the common methods, and also, and more particularly, of other methods, 
greatly preferable to them in practice if I am not mistaken, by means of 



PREFACE. Xvii 

which we pass from the known elements to the phenomena; the former treat 
of many most important problems which prepare the way to inverse pro- 
cesses. Since these very phenomena result from a certain artificial and intri- 
cate complication of the elements, the nature of this texture must be thor- 
oughly examined before we can undertake with hope of success to disentangle 
the threads and to resolve the fabric into its constituent parts. Accordingly, 
in the First Book, the means and appliances are provided, by means of which, 
in ,the second, this difficult task is accomplished ; the chief part of the labor, 
therefore, consists in this, that these means should be properly collected to- 
gether, should be suitably arranged, and directed to the proposed end. 

The more important problems are, for the most part, illustrated by appro- 
priate examples, taken, wherever it was possible, from actual observations. 
In this way not only is the efl&cacy of the methods more fully established 
and their use more clearly shown, but also, care, I hope, has been taken that 
inexperienced computers should not be deterred from the study of these sub- 
jects, which undoubtedly constitute the richest and most attractive part of 
theoretical astronomy. 

GoTTiNGEN, March. 28, 1809. 



FIRST BOOK. 



GENERAL RELATIONS BETWEEN THOSE QUANTITIES BY WHICH THE 
MOTIONS OF HEAVENLY BODIES ABOUT THE SUN ARE DEFINED. 



FIRST SECTION. 

KELATIONS PERTAINING SIMPLY TO POSITION IN THE ORBIT. 



1. 

In this work we shall consider the motions of the heavenly bodies so far only 
as they are controlled by the attractive force of the sun. All the secondary 
planets are therefore excluded from our plan, the perturbations which the 
primary planets exert upon each other are excluded, as is also all motion of 
rotation. "We regard the moving bodies themselves as mathematical points, and 
we assume that all motions are performed in obedience to the following laws, 
which are to be received as the basis of all discussion in this work. 

I. The motion of every heavenly body takes place in the same fixed 
plane in which the centre of the sun is situated. 

II. The path described by a body is a conic section having its focus in the 
centre of the sun. 

III. The motion in this path is such that the areas of the spaces described 
about the sun in different intervals of time are proportional to those intervals. 
Accordingly, if the times and spaces are expressed in numbers, any space what- 
ever divided by the time in which it is described gives a constant quotient. 

1 



2 RELATIONS PERTAINING SBITLY [BoOK I. 

IV. For different bodies moving about the sun, the squares of these quotients 
are in the compound ratio of the parameters of their orbits, and of the sum of the 
masses of the sun and the moving bodies. 

Denoting, therefore, the parameter of the orbit in which the body moves by 
2ji9, the mass of this body by f^ (the mass of the sun being put =1), the area it 

describes about the sun in the time t by ^(/, then ^ , ^q j_ n will be a constant 

for all heavenly bodies. Since then it is of no importance which body we use 
for determining this number, we will derive it from the motion of the earth, the 
mean distance of which from the sun we shall adopt for the unit of distance ; the 
mean solar day will always be our unit of time. Denoting, moreover, by tt the 
ratio of the circumference of the circle to the diameter, the area of the entire 
ellipse described by the earth will evidently be n \J p, which must therefore be 
put = ^^, if by ^ is understood the sidereal year; whence, our constant becomes 

— — — — - . In order to ascertain the numerical value of this constant, here- 

after to be denoted by k, we will put, according to the latest determination, the 

sidereal year or if = 365.2563835, the mass of the earth, or fi:^ =: 

0.0000028192, whence results 

log27r 0.7981798684 

Compl. log^ 7.4374021852 

Compl. log. v/(14-ii^) . . . 9.9999993878 

log /f 8.2355814414 

k= 0.01720209895. 

2. 

The laws above stated differ from those discovered by our own Kepler 
in no other respect than this, that they are given in a form applicable to all kinds 
of conic sections, and that the action of the moving body on the sun, on which 
depends the factor y'(l-)-^), is taken into account. If we regard these laws as 
phenomena derived from innumerable and indubitable observations, geometry 
shows what action ought in consequence to be exerted upon bodies moving about 



Sect. 1.] to position in the orbit. 3 

the sun, in order that these phenomena may be continually produced. In this 
way it is found that the action of the sun upon the bodies moving about it is 
exerted just as if an attractive force, the intensity of which is reciprocally 
proportional to the square of the distance, should urge the bodies towards the 
centre of the sun. If now, on the other hand, we set out with the assumption of 
such an attractive force, the phenomena are deduced from it as necessary 
consequences. It is sufficient here merely to have recited these laws, the con- 
nection of which with the principle of gravitation it will be the less necessary to 
dwell upon in this place, since several authors subsequently to the eminent 
Newton have treated this subject, and among them the illustrious La Place, in 
that most perfect work the Mecanique Celeste, in such a manner as to leave 
nothing further to be desired. 

3. 

Inquiries into the motions of the heavenly bodies, so far as they take place in 
conic sections, by no means demand a complete theory of this class of curves ; 
but a single general equation rather, on which all others can be based, will answer 
our purpose. And it appears to be particularly advantageous to select that one 
to which, while investigating the curve described according to the law of attrac- 
tion, we are conducted as a characteristic equation. If we determine any place 
of a body in its orbit by the distances x, y, from two right lines drawn in the 
plane of the orbit intersecting each other at right angles in the centre of the 
sun, that is, in one of the foci of the curve, and further, if we denote the distance 
of the body from the sun by r (always positive), we shall have between r, x, p, 
the linear equation r-[- a :«;-)- /3^=r/, in which «, (3, / represent constant quan- 
tities, / being from the nature of the case always positive. By changing the 
position of the right lines to which x,i/, are referred, this position being essentially 
arbitrary, provided only the lines continue to intersect each other at right angles, 
the form of the equation and also the value of / will not be changed, but the 
values of a and /i will vary, and it is plain that the position may be so determined 
that (i shall become = 0, and «, at least, not negative. In this way by putting for 
«, ?'; respectively e.p, our equation takes the form r-\-ex=^p. The right line to 



4 RELATIONS PERTAINING SBIPLY BoOK I. 

which the distances y are referred in this case, is called the line of ajysides, p is the 
semi-parameter, e the eccentriciti/ ; finally the conic section is distinguished by the 
name of ellijjse, parabola, or hyperlola, according as e is less than unity, equal to 
unity, or greater than unity. 

It is readily perceived that the position of the line of apsides would be 
fully determined by the conditions mentioned, with the exception of the single 
case where both a and /5 were = ; in which case r is always =^?, whatever the 
right lines to which x, y, are referred. Accordingly, since we have 6 = 0, the 
curve (which will be a circle) is according to our definition to be assigned to 
the class of ellipses, but it has this peculiarity, that the position of the apsides 
remains wholly arbitrary, if indeed we choose to extend that idea to such a case. 



4. 

Instead of the distance x let us introduce the angle v, contained between the 
line of apsides and a straight line drawn from the sun to the place of the body 
{the radius vector), and this angle may commence at that part of the line of apsides 
at which the distances x are positive, and may be supposed to increase in the 
direction of the motion of the body. In this way we have x = r cos v, and thus 
our formula becomes r = :r-r , from which immediately result the following 

1 -|- e cos w ' -^ ° 

conclusions : — 

I. For v^ 0, the value of the radius vector r becomes a minimum, that is, 
= l^T^ • this j)oint is called the perihelion. 

II. For opposite values of v, there are corresponding equal values of r ; con- 
sequently the line of apsides divides the conic section into two equal parts. 

III. In the ellipse, v increases continuously from y = 0, until it attains its 
maximum value, y^^, in a^M?'ra, corresponding to y = 180°; after aphelion, it 
decreases in the same manner as it had increased, until it reaches the perihelion, 
corresponding to v:=^ 360°. That portion of the line of apsides terminated at one 
extremity by the perihelion and at the other by the aphelion is called the major 



Sect. 1.] to position in the orbit. 5 

axis ; hence the semi-axis major, called also the mean distance, = y^ — ; the dis- 
tance of the middle point of the axis {the centre of the .ellipse) from the focus will 
be ^ = ea, denoting by a the semi-axis major. 

IV. On the other hand, the aphelion in its proper sense is wanting in the 
parabola, but r is increased indefinitely as v approaches -|- 180°, or — 180°. For 
V = + 180° the value of r becomes infinite, which shows that the curve is not cut 
by the line of apsides at a point opposite the perihelion. Wherefore, we cannot, 
with strict propriety of language, speak of the major axis or of the centre of the 
curve ; but by an extension of the formulas found in the ellipse, according to the 
established usage of analysis, an infinite value is assigned to the major axis, and 
the centre of the curve is placed at an infinite distance from the focus. 

Y. In the hyperbola, lastly, v is confined within still narrower limits, in fact 
between v= — (180° — \\i), and y = -|-(180° — ^i), denoting by x^ the angle of 

which the cosine =-. For whilst v approaches these limits, r increases to 

infinity ; if, in fact, one of these two limits should be taken for v, the value of r 
would result infinite, which shows that the hyperbola is not cut at all by a right 
line inclined to the line of apsides above or below by an angle 180° — if. For 
the values thus excluded, that is to say, from 180° — i// to 180°-[-^j our formula 
assigns to r a negative value. The right line inclined by such an angle to the 
line of apsides does not indeed cut the hyperbola, but if produced reversely, 
meets the other branch of the hyperbola, which, as is known, is wholly separ 
rated from the first branch and is convex towards that focus, in which the sun is 
situated. But in our investigation, which, as we have already said, rests upon the 
assumption that r is taken positive, we shall pay no regard to that other branch 
of the hyperbola in which no heavenly body could move, except one on which 
the sun should, according to the same laws, exert not an attractive but a repulsive 
force. Accordingly, the aphelion does not exist, properly speaking, in the hyper- 
bola also ; that point of the reverse branch which lies in the line of apsides, 
and which corresponds to the values y = 180°, r = — ^~ii might be consid- 
ered as analogous to the aphelion. If now, we choose after the manner of the 



6 KELATIONS PERTAINING SBIPLY [BoOK I. 

ellipse to call the value of the expression _^ , even here where it becomes 
negative, the semi-axis major of the hyperbola, then this quantity indicates 
the distance of the point just mentioned from the perihelion, and at the 
same time the position opposite to that which occurs in the ellipse. In the 
same way ^ , that is, the distance from the focus to the middle point between 
these two points (the centre of the hyperbola), here obtains a negative value on 
account of its opposite direction. 



We call the angle v the true anomaly of the moving body, which, in the 
parabola is confined within the limits — 180° and -{- 180°, in the hyperbola 
between — (180° — i/^) and -|- (180° — t/^), but which in the ellipse runs through 
the whole circle in periods constantly renewed. Hitherto, the greater number of 
astronomers have been accustomed to count the true anomaly in the ellipse not 
from the perihelion but from the aphelion, contrary to the analogy of the parabola 
and hyperbola, where, as the aphelion is wanting, it is necessary to begin from the 
perihelion : we have the less hesitation in restoring the analogy among all classes 
of conic sections, that the most recent French astronomers have by their example 
led the way. 

It is frequently expedient to change a little the form of the expression 

r = r—r-^ : the following forms will be especially observed : — 

1 -|- e cos v' " r ./ 

y— P P 

1 _[- e — 2 e sin^ ^v 1 — e-\-2e cos^ ^ v 

^_ P 



(1 -|- e) cos^ \v -\-{\ — e) sin'-^ ^ v ' 

Accordingly, we have in the parabola 

P ^ 

^~2cos2it;' 

in the hyperbola the following expression is particularly convenient, 

^_ -Pcost/; ^ 

2cos^(i>-l-i/;)cos^(u — i/))* 



kJiiCT. 1.] 



TO POSITION IN THE ORBIT. 



6. 

Let us proceed now to the comparison of the motion with the time. Putting, 
as in Art. 1, the space described about the sun in the time t=hg, the mass of the 
moving body = ^, tliat of the sun being taken = 1, we have g =zJi:t\^p\j {l-\- ^a). 
The differential of the space = hrrdv, from which there results Jtt\JpsJ {1 ^ ^) 
=:frrdiV, this integral being so taken that it will vanish for ^ = 0. This integra- 
tion must be treated differently for different kinds of conic sections, on which 
account, we shall now consider each kind separately, beginning with the ELLIPSE. 

Since r is determined from v by means of a fraction, the denominator of which 
consists of two terms, we will remove this inconvenience by the introduction of a 
new quantity in the place of v. For this purpose we will put tan ^v ^ y~T~ =^ 
tan h E, by which the last formula for r in the preceding article gives 

^ — (l^e)cos^v—^\l-\-e^l—e} — l-ee^^ 6 COS ^ j . 

Moreover we have — rj-m = — ¥t-\/ t-t—) and consequently dv= —M. r: 

hence 



'■'■d^=v-^=^i(i-"-^)<'^' 



and integrating, 



Mslp^{l-^lJi)= P^ 3 (^— g sin ^)-f Constant. 
Accordingly, if we place the beginning of the time at the perihelion passage, where 



^) = 0, ^= 0, and thus constant = 0, we shall have, by reason of ^_^^ ■= a, 

J 

In this equation the auxiliary angle E, which is called the eccentric anomaly^ 

must be expressed in parts of the radius. This angle, however, may be retained 

in degrees, etc., if e sin E and — ^ 3 are also expressed in the same manner ; 

these quantities will be expressed in seconds of arc if they are multiplied by the 



8 KELATIONS PERTAINING SBIPLY [BoOK I. 

number 206264.81. We can dispense with the multiphcation by the last quan- 
tity, if we employ directly the quantity h expressed in seconds, and thus put, 
instead of the value before given, Ti = 3548".18761, of which the logarithm = 
3.5500065746. The quantity — 3 expressed in this manner is called the 

mean anomaly, which therefore increases in the ratio of the time, and indeed every 
day by the increment , , called the mean daily motion. "We shall denote 

the mean anomaly by M. 

7: 
Thus, then, at the perihelion, the true anomaly, the eccentric anomaly, and the 
mean anomaly are = ; after that, the true anomaly increasing, the eccentric 
and mean are augmented also, but in such a way that the eccentric continues to 
be less than the true, and the mean less than the eccentric up to the aphelion, 
where all three become at the same time = 180°; but from this point to 
the perihelion, the eccentric is always greater than the true, and the mean 
greater than the eccentric, until in the perihelion all three become = 360°, or, 
which amounts to the same thing, all are again = 0. And, in general, it is 
evident that if the eccentric U and the mean M answer to the true anomaly v, 
then the eccentric 360° — U and the mean 360° — M correspond to the true 
360° — V. The difference between the true and mean anomalies, v — J^ is called 
the equation of the centre, which, consequently, is positive from the perihelion 
to the aphelion, is negative from the aphelion to the perihelion, and at the 
perihelion and aphelion vanishes. Since, therefore, v and 31 run through an 
entire circle from to 360° in the same time, the time of a single revolution, 
which is also called the periodic time, is obtained, expressed in days, by dividing 

360° by the mean daily motion v ^ 7~ , from which it is apparent, that for dif 

a- 

ferent bodies revolving about the sun, the squares of the periodic times are pro- 
portional to the cubes of the mean distances, so far as the masses of the bodies, 
or rather the inequality of their masses, can be neglected. 



Sect. 1.] to position in the orbit. 9 

8. 
Let us now collect together those relations between the anomalies and the 
radius vector which deserve particular attention, the derivation of which will 
present no difficulties to any one moderately skilled in trigonometrical analysis. 
Greater elegance is attained in most of these formulas by introducing in the 
place of e the angle the sine of which = e. This angle being denoted by 9, we 
have 

y/(l_gg) = cos9), y/(l-[-e)=:cos(45° — i9))v/2, 

^(1 — e) = cos(45°+f9)v/2, y/l^==tan(45° — ^9), 

y/ (1 _f_ e) _^ ^(1 _ e) = 2 cos 1 9, sj {\^ e) — ^{\ — e) = 'lmx\<^. 

The following are the principal relations between a, p, r, e, cp, v, E, M. 

I. p = a cos^ (p 

1 -j- e cos « 

III. r = a{l — ecosE) 

l\r ^^ E7 cosv-4-e cos^ — e 

IV. cos^ = --| — =c_ or cos ZJ = :; ^ 

1 -f- e cos w ' 1 — e cos Mi 

V. smiE=J^(l — cosE)=smivJy-^~^^-^ 

' ^ ^ V 1 -|- e cos s; 

= Sin i . ^^^^^ ^ sin ^ . y/^^^ 

VL cos^^=:v/^l + cos^)=cos^.y/j-li^^ 

= cos ^y 1/— ^-i^ = COS hvJ—rr- — ^ 

Vn. tani^=tan^z;tan(45°— ^9) 

VIII. sin ^^ "^"""'""^ ^ '"''"'' 
p a cos qo 

IX. rcosz; = a(cos^— e) = 2«cos(^^+i9) + 45°)cos(^J^— ig) — 45°) 

X. sin h{v — ^) = sin J 9 sin ^; w - = sin J g) sin ^ i/ - 

XL sin i (z; -}- -^) = cos i 9 sin tJ i/ - =: cos J 9 sin J' w- 

XIL M=E—esmE. 

2 



10 RELATIONS PERTAINING SIMPLY [BoOK I. 



If a perpendicular let fall from any point whatever of the ellipse upon the 
line of apsides is extended in the opposite direction until it meets the circle 
described with the radius a about the centre of the ellipse, then the inclination to 
the line of apsides of that radius which corresponds to the point of intersection 
(understood in the same way as above, in the case of the true anomaly), will 
be equal to the eccentric anomaly, as is inferred without difficulty from equation 
IX. of the preceding article. Further, it is evident that r sin v is the distance of 
any point of the ellipse from the line of apsides, which, since by equation VIII. it 
= a cos 9 sin E, will be greatest for E^=. 90°, that is in the centre of the ellipse. 
This greatest distance, which = a cos y ^=^ -^ ■=.'^ap, is called the semi-axis minor. 
In the focus of the ellipse, that is for y = 90°, this distance is evidently ■=p, or 
equal the semi-parameter. 

10. 

The equations of article 8 comprise all that is requisite for the computation 
of the eccentric and mean anomalies from the true, or of the eccentric and true 
from the mean. Formula VII. is commonly employed for deriving the eccentric 
from the true ; nevertheless it is for the most part preferable to make use of 
equation X. for this purpose, especially when the eccentricity is not too great, in 
which case E can be computed with greater accuracy by means of X. than of 
VII. Moreover, if X. is employed, the logarithm of sine E required in XII. is 
had immediately by means of VIII. : if VII. were used, it would be neces- 
sary to take it out from the tables; if, therefore, this logarithm is also taken 
from the tables in the latter method, a proof is at once obtained that the calcula- 
tion has been correctly made. Tests and proofs of this sort are always to be 
highly valued, and therefore it will be an object of constant attention with us to 
provide for them in all the methods delivered in this work, where indeed it can 
be conveniently done. We annex an example completely calculated as a more 
perfect illustration. 



Sect. 1.] to position in the orbit. 11 

Given V = 310° 55' 29^64, 9 = 14° 12' r.87, log r = 0.3307640 ; p, a, U, M, 
are required. 

log sin 9 .... 9.3897262 
log cosy .... 9.8162877 



9.2060139 whence e cos y = 0.1606993 



log (1 -I- e cos?;). . 0.0647197 
logr 0.3307640 

\ogp 0.3954837 

logcos> .... 9.9730448 

log a 0.4224389 

log sine; .... 9.8782740 ;2* 

logi/f .... 0.0323598.5 

9.8459141.5W 
log sin ^ 9 ... 9.0920395 

log sin ^(y — ^) . 8.9379536.5^, hence ^(y — ^) = — 4° 58' 22".94; 
y — ^ = — 9° 56' 45".88 ; E= 320° 52' 15".52. 
Further, we have 

■ „ „^„w^^^ Calculation of log sin E by formula Vm. 

loge . . . . 9.3897262 ^ 

1 oAcoc.( o KQT^/ion log-smy .... 9.8135543w 

log 206264.8 . 5.3144251 ^ p 

log e in seconds 4.7041513 log cos 9 9.9865224 



logsin^. . . 9.8000767w logsin^ 9.8000767w 

4.5042280 n, hence e sin ^ in seconds = 31932".14 = 8° 52' 
12".14 ; and M= 329° 44' 27".66. 

The computation of E by formula VII. would be as follows : — 

^?; = 155°27'44".82 log tan ^y .... 9.6594579?2 

] 45° — l9) = 37"53'59".065 log tan (45° — i 9) . 9.8912427 

log tan ^^ . . . . 9.5507006 « 
whence hE= 160''26'7".76, and E= 320°52'15".52, as above. 

* The letter « affixed to a logarithm signifies that the number corresponding to it is negative. 



12 RELATIONS PERTAINING SBIPLY [BoOK I. 

11. 

The inverse problem, celebrated under the title of Kepler's problem, that of 
finding the true anomaly and the radius vector from the mean anomaly, is much 
more frequently used. Astronomers are in the habit of putting the equation of 
the centre in the form of an infinite series proceeding according to the sines of the 
angles M, 2M, SM, etc., each one of the coefficients of these sines being a series 
extending to infinity according to the powers of the eccentricity. We have con- 
sidered it the less necessary to dwell upon this formula for the equation of the 
centre, which several authors have developed, because, in our opinion, it is by 
no means so well suited to practical use, especially should the eccentricity not be 
very small, as the indirect method, which, therefore, we will explain somewhat 
more at leng-th in that form which appears to us most convenient. 

Equation XII., U = M-\- e sin H, which is to be referred to the class of tran- 
scendental equations, and admits of no solution by means of direct and complete 
methods, must be solved by trial, beginning with any approximate value of jC, which 
is corrected by suitable methods repeated often enough to satisfy the preceding 
equation, that is, either with all the accuracy the tables of sines admit, or at least 
with sufficient accuracy for the end in view. If now, these corrections are intro- 
duced, not at random, but according to a safe and established rule, there is scarcely 
any essential distinction between such an indirect method and the solution by 
series, except that in the former the first value of the unknown quantity is in a 
measure arbitrary, which is rather to be considered an advantage since a value 
suitably chosen allows the corrections to be made with remarkable rapidity. Let 
us suppose £ to be an approximate value of U, and z expressed in seconds the cor- 
rection to be added to it, of such a value as will satisfy our equation JS=t-\-x. 
Let e sin e, in seconds, be computed by logarithms, and when this is done, let the 
change of the log sin e for the change of 1" in e itself be taken from the tables ; 
and also the variation of log e sin e for the change of a unit in the number e sin s ; 
let these changes, without regard to signs, be respectively I, fi, in which it is 
hardly necessary to remark that both logarithms are presumed to contain an 
equal number of decimals. Now, if e approaches so near the correct value of ^ 



Sect. 1.] to position in the orbit. 13 

that the changes of the logarithm of the sine from s to e-\-x, and the changes of 
the logarithm of the number from e sin e to e sin (s -j- x), can be regarded as 
uniform, we may evidently put 

7 X 

e sin (s -]- rr) = e sin £ + — , 

the upper sign belonging to the first and fourth quadrants, and the lower to the 
second and third. Whence, since 

e -\~x = M-\- e sin (e -f- x), we have x = -~ {^-\- & sin e — e), 

and the correct value of ^,or 

£ -|-a; = i!^f -]- ^sine+^=rj(7{f-|- esine — e), 

the signs being determined by the above-mentioned condition. 

Finally, it is readily perceived that we have, without regard to the signs, 
li-.X^X-.e cos e, and therefore always ^^^, whence we infer that in the first and 
last quadrant M-\- e sin e lies between e and e -\- x, and in the second and third, 
e-^x between t and M-\-esin e, which rule dispenses with paying attention to the 
signs. If the assumed value e differs too much from the truth to render the fore- 
going considerations admissible, at least a much more suitable value will be found 
by this method, with which the same operation can be repeated, once, or several 
times if it should appear necessary. It is very apparent, that if the difierence 
of the first value £ from the truth is regarded as a quantity of the first order, the 
error of the new value would be referred to the second order, and if the operation 
were further repeated, it would be reduced to the fourth order, the eighth order, 
etc. Moreover, the less the eccentricity, the more rapidly will the successive 
corrections converge. 

12. 

The approximate value of ^, with which to begin the calculation, will, in most 
cases, be obvious enough, particularly where the problem is to be solved for 
several values of M of which some have been already found. In the absence 
of other helps, it is at least evident that U must fall between M and M±e, (the 
eccentricity e being expressed in seconds, and the upper sign being used in the 



14 RELATIONS rERTiVLNTNG SIMPLY [BoOK 1. 

first and second quadrants, the lower in the third and fourth), wherefore, either 
3f, or its value increased or diminished by any estimate whatever, can be taken 
for the first value of E. It is hardly necessary to observe, that the first calcu- 
lation, when it is commenced with a value having no pretension to accuracy, does 
not require to be strictly exact, and that the smaller tables * are abundantly suffi- 
cient. Moreover, for the sake of convenience, the values selected for £ should be 
such that their sines can be taken from the tables without interpolation ; as, for 
example, values to minutes or exact tens of seconds, according as the tables 
used proceed by differences of minutes or tens of seconds. Every one will be 
able to determine without assistance the modifications these precepts undergo if 
the angles are expressed according to the new decimal division. 

13. 

Example. — Let the eccentricity be the same as in article 10. J!f=332°28' 
54".77. There the log e in seconds is 4.7041513, therefore e= 50600" = 14° 3'20". 
Now since E here must be less than M, let us in the first calculation put e ==: 326°, 
then we have by the smaller tables 

log sine 9.74756;?, Changeforr ... 19, whence a = 0.32. 

log; c in seconds . . 4.70415 



4.45171w; 

hence esin£= 28295 = 7°5135. change of logarithm for a unit of the table which is here 

MA- e sin £ 324 37 20 ^^^'^ *° ■"* ^^'^°^^^ . . . I6; whence /i=1.6. 

differing from £ .... 1 22 40 = 4960". Hence, 
?4| X 4960" = 1240" = 20'40". 

Wherefore, the corrected value of E becomes 324° 37' 20"— 20' 40"= 324° 16'40", 
with which we repeat the calculation, making use of larger tables. 

log sin £ .... 9.7663058;e 2, = 29.25 

loge 4.7041513 

4.4704571 w /* = 147 

* Such as those which the illusti-ious Lalaxde furnisher!. 



Sect. 1.] to position in the orbit. 15 

e sin e = — 29543'a8 = — 8° 12' 23'a8 

M-\-esme .... 324 1631.59 

differing from e . * . 8 .41. 

X 29 25 . 

This difference being multiplied by -^^3= „„g gives 2''.09, whence, finally, the 

corrected value of ^ = 324°16'3r.59 — 2".09 = 324°16'29''.50, which is exact 
within 0''.01. 



14. 

The equations of article 8 furnish several methods for deriving the true 
anomaly and the radius vector from the eccentric anomaly, the best of which we 
will explain. 

I. By the common method v is determined by equation VII., and afterwards 
r by equation II.; the example of the preceding article treated in this way 
is as follows, retaining for jt? the value given in article 10. 

i^=162°8'14".75 loge 9.3897262 

log tan ^^. . . . 9.5082198n log cosy .... 9.8496597 
log tan (45°— i 9) . 9.8912427 9.2393859 

logtan^y . . . . 9.6169771 w ecosy =0.1735345 

iv = 157° 30'4r.50 l^ 0.3954837 

t' = 315 123.00 log (1-f-e cost'). . 0.0694959 

"b^T 0.3259878. 

II. The following method is shorter if several places are to be computed, 
for which the constant logarithms of the quantities ^ a{l-\-e), )J a{l — e) should 
be computed once for all. By equations V. and VI. we have 

smiv^r^=smiJS )^ a{l-\-e) 

cos i y \/ r = cos ^ -E' \/ a{l — e) 
from which i v and log \J r are easily determined. It is true in general that if we 
have Psin Q = A, P cos Q=zB, ^ is obtained by means of the formula tan 
Q = ^, and then P by this, P = ^, or by P = ^^ : it is preferable to use 



16 



KELATIONS PERTAINING SIMPLY 



[Book 1. 



the former when sin Q is greater than cos Q ; the latter when cos Q is greater than 
sin Q. Commonly, the problems in which equations of this kind occur (such as 
present themselves most frequently in this work), involve the condition that P 
should be a positive quantity ; in this case, the doubt whether Q should be taken 
between and 180°, or between 180° and 360°, is at once removed. But if such 
a condition does not exist, this decision is left to our judgment. 
We have in our example e = 0.2453162. 

log sin i^ . . . 9.4867632 log cos ^^ . . . 9.9785434w 

0.2588593 



logv/a(14-e) . 

Hence 

log sm^v \J r . 
log cos IvsJ r . 
log cos iv . . 



log y/«(1 — e) 



0.1501020. 



9.7456225 
0.1286454 
9.9656515^ 



J 



whence, log tan hv= 9.6169771m 

iy = 157°30'4r.50 
z; = 315 123.00 



\ogsJr .... 0.1629939 
losr 0.3259878 



III. To these methods we add a third which is almc^t equally easy and expe- 
ditious, and is much to be preferred to the former if the greatest accuracy should 
be required. Thus, ris first determined by means of equation III, and after that, 
V by X. Below is our example treated in this manner. 

loa;e 9.3897262 



cos JE 



9.9094637 



log sin ^ . . . 

log v'(l — ecosE) 



9.7663366« 
9.9517744 



ecos-E'= 



9.2991899 
0.1991544 



log sin h (p 



9.8145622^ 
9.0920395 



log a . . . . 
log(l — cgobE) 



0.4224389 
9.9035488 



logr 0.3259877 



\ogsmh{v — E) . . 8.9066017« 

j(y_^)=_4°3r 33-24 

v — E r^ — 9 15 6.48 

?;c=315 123.02 



Formula VIII., or XI., is very convenient for verifying the calculation, par- 
ticularly if V and r have been determined by the third method. Thus ; 



Sect. 1.] to position in the orbit. 17 

log^sin^ . . . 9.8627878^ logsin^V^ . . . 9.8145622?^ 
log cos 9 .... 9.9865224 logcos^y . . . . 9.9966567 

9.8493102^ 9.8112189W 

logsiny . . . , 9.8493102» log sin | (y + ^) . . 9.8112189w 

15. 

Since, as we have seen, the mean anomaly M is completely determined by 
means of v and 9, in the same manner as e^ by Jlf and 9, it is evident, that if all 
these quantities are regarded as variable together, an equation of condition ought 
to exist between their differential variations, the investigation of which will not 
be superfluous. By differentiating first, equation YIL, article 8, we obtain 

^E ^ dy _ 

sin^ sinu cosgj' 

by differentiating likewise equation XII., it becomes 

dif=(l — eco%E)diE — sin ^ cos 9 d 9. 
If we eliminate d^from these differential equations we have 

dJf=: ^-^ '-^v — (sm^cosg)H ^^ ^jd^j 

sm V \ ' ' cos go / ' 

or by substituting for sin E,\ — e cos E, their values from equations VIII., III., 
dJf:=-^:^de>- "^'-+^>r" dy, 

a a cos QD a a cos** g) ' 

or lastly, if we express both coefficients by means of v and 9 only, 

(^j/__ cos^9 ^„j (2-j-ecost))sinycos^qp ^ 

(l-j-ficosw)^ (l-}-ecosv)^ '' 

Inversely, if we consider z; as a function of the quantities M, cp, the equation has 
this form : — 

^^^ ggcosg, ^^ , (2 + .cosz.)smz; ^ 

rr ' cos qp ^ ' 

or by introducing E instead o{ v 

dv = ^^^dM-{-^^^i2—ecosE—ee)smEd(p. 



18 RELATIONS PERTAINING SDIPLY [BoOK I. 

16. 

The radius vector r is not fully determined by v and 9, or by M and g), but 
depends, besides these, upon p or a; its differential, therefore, ■will consist of three 
parts. By differentiating equation IT. of article 8, we obtain 
dr dp 



u. / VI t/ I c 0111 V 1 

— =.-i--\-—-. -dy- 

v> -n • 1 -4— p. pn^ 7J 



1 -|- e cos V 1 -[- e cos v ' 

By putting here 

— = 2 tan cp d CO 

pa ' ^ 

(which follows from the differentiation of equation I.), and expressing, in con- 
formity with the preceding article, d y by means of d i!f and d (p, we have, after 
making the proper reductions, 

dr da , a . . i i,t a ^ 

— = h- tan wsmvaM cos cp cos vdw, 

r a ' r r ' ' 

dr = - da-l-«tan9)sin^>dJf — a cos 9 cos yd 9. 

Finally, these formulas, as well as those which we developed in the preceding 
article, rest upon the supposition that v, cp, and M, or rather dv, d 9, and d M, 
are expressed in parts of the radius. If, therefore, we choose to express the vari- 
ations of the angles v, (p, and M, in seconds, we must either divide those parts of 
the formulas which contain dv,d(p,or dM,hy 206264.8, or multiply those which 
contain d r, dp, d a, by the same number. Consequently, the formulas of the pre- 
ceding article, which in this respect are homogeneous, will require no change. 

17. 

It will be satisfactory to add a few words concerning the investigation of the 
greatest equation of the centre. In the first place, it is evident in itself that the dif- 
ference between the eccentric and mean anomaly is a maximum for ^= 90°, 
where it becomes = e (expressed in degrees, etc.) ; the radius vector at this point 
= a, whence v = 90° -|- y, and thus the whole equation of the centre = 9) -]- e, 



Sect. 1.] to position m the orbit. 19 

which, nevertheless, is not a maximum here, since the difference between v and 
U may still increase beyond cp. This last difference becomes a maximum for 

d (y ^) = or for dv = dU, where the eccentricity is clearly to be regarded 

as constant. "With this assumption, since in general 

dv dU 

sin V sin U ' 

it is evident that we should have sin ^J = sin ^ at that point where the difference 
between v and ^ is a maximum ; whence we have by equations VIIL, HI, 

r^=acos<p, e cos^= 1 — cos 9, or cos^^-f"*^^ ^ 9- 
In like manner cos z; = — tan i cp is found, for which reason it will follow * that 

V = 90° -f- arc sin tan ^9, ^ = 90° — arc sin tan | (f ; 
hence again 

sin^ = v/ (1 — tan^ i (p) = ^^, 

SO that the whole equation of the centre at this point becomes 

2 arc sin tan i 9 -|- 2 sin ^ 9) \/ cos (f, 
the second term being expressed in degrees, etc. At that point, finally, where 
the whole equation of the centre is a maximum, we must have dy = dil/^ and 
so according to article 15, r=a^ cos (p ; hence we have 

1 COS^ op rt 1 \/ COS CD 1 COS (Y) tajQ i QD 

cose; = ^, cos-E^== "^ -= .., , , ^ = , , , , 

e ' e e {1 -\- ^ cos cp) l-j-ycosg;' 

by which formula U can be determined with the greatest accuracy. U being 
found, we shall have, by equations X.,XII., 

equation of the centre = 2 arc sin ^^"/ ^ ^^" 1- e sin U. 

\J COS(p 

"We do not delay here for an expression of the greatest equation of the centre by 
means of a series proceeding according to the powers of the eccentricities, which 
several authors have given. As an example, we annex a view of the three 
maxima which we have been considering, for Juno, of which the eccentricity, 
according to the latest elements, is assumed = 0.2554996. 

* It is not necessary to consider those maxima which lie between the aphelion and perihelion, 
because they evidently differ in the signs only from those which are situated between the perihelion and 
aphelion. 



20 



RELATIONS PERTAINING SIMPLY 



[Book I. 



Maximum. 


E 


E—M 


v—E 


v—M 


E—M 
v—E 
v — M 


90° 0' 0" 
82 32 9 
86 14 40 


14° 38' 20".57 
14 30 54 .01 
14 36 27 .39 


14°48'11".48 
14 55 41 .79 
14 53 49 .57 


29°26'32".05 
29 26 35 .80 
29 30 16 .96 



18. 

In the PARABOLA, the eccentric anomaly, the mean anomaly, and the mean 
motion, become = ; here therefore these ideas cannot aid in the comparison of 
the motion with the time. In the parabola, however, there is no necessity for an 
auxiliary angle in integrating rrdv; for we have 



4cos^ 



2 cos^ 4 V 



' PP (1 -|-tan^ hv) dtan hv; 



and thus, 

frr^v = hpp (tan hv -\- k tan^ ^ ^) + Constant. 
If the time is supposed to commence with the perihelion passage, the Constant 
= ; therefore we have 

tani.+ Han^i. = ^-^^^^^+^, 

by means of which formula, t may be derived from v, and v from t, when p and 
(Jb are known. In the parabolic elements it is usual, instead of p, to make use of 
the radius vector at the perihelion, which is h p, and to neglect entirely the mass 
(I. It will scarcely ever be possible to determine the mass of a body, the orbit of 
which is computed as a parabola ; and indeed all comets appear, according to the 
best and most recent observations, to have so little density and mass, that the 
latter can be considered insensible and be safely neglected. 



19. 

The solution of the problem, from the true anomaly to find the time, and, in 
a still greater degree, the solution of the inverse problem, can be greatly abbrevi- 
ated by means of an auxiliary table, such as is found in many astronomical works. 



Sect. 1.] to position in the orbit. 21 

But the Barkerian is by far the most convenient, and is also annexed to the 
admirable work of the celebrated Olbeks, [Ahlmndlung uber die leichteste und 
leqiiemste Methode die Bahn eines Cometen su lerechnen : Weimar, 1797.) It contains, 
under the title of the mean motion, the value of the expression 75 tan ^y-]- 25 
tan^ ^ V, for all true anomalies for every five minutes from to 180°. If 
therefore the time corresponding to the true anomaly v is required, it will be 

necessary to divide the mean motion, taken from the table with the arguments, 

150 h • 

by — ^, which quantity is called the mean daily motion; if on the contrary the 

true anomaly is to be computed from the time, the latter expressed in days will 
be multiplied by — ^, in order to get the mean motion, with which the correspond- 

ing anomaly may be taken from the table. It is further evident that the same 
mean motion and time taken negatively correspond tp the negative value of the v ; 
the same table therefore answers equally for negative and positive anomalies. If 
in the place of />,we prefer to use the perihelion distance \'p-=zq, the mean daily 

^ v/2812 5 
motion is expressed by ^ ^ ' , in which the constant factor ^y/ 2812,5 =^ 

0.912279061, and its logarithm is 9.9601277069. The anomaly v being found, 
the radius vector will be determined by means of the formula already given. 



20. 

By the differentiation of the equation 

tan \v-\-l tan^ ^ y = 2 t'kp~^, 
if all the quantities v, t,p, are regarded as variable, we have 






22 RELATIONS PERTAIMNG SBIPLY [BoOK I. 

If the variations of the anomaly v are wanted in seconds, both parts also of 
^v must be expressed in this manner, that is, it is necessary to take for k the value 
3548".188 given in article 6. If, moreover, ^p=q is introduced instead of p, the 
formula wiU have the following form : 

rr rrsjzq ^' 

in which are to be used the constant logarithms 

log ks/2 = 3.7005215724, log^/c^i = 3.8766128315. 
Moreover the differentiation of the equation 

furnishes 

— = —-I- tan ivdiV, 

or by expressing dv by means of d^ and dp, 

Ar /I Zht\a.n\v\ ■, j^lc\jpian\v ■, , 

r \p %rr^p /-'"'" ^.^ 

By substituting for i its value in v, the coefficient of dp is changed into 

1 _ 3^Un^ _ ^jan^ _ 1 / ^^^, .^^ 



but the coefl&cient of d^ becomes — t—* From this there results 



or if we introduce q for p 



z 1 ;i , ^ sin w , , 

d r = i cos z; d j» -[" "w — " ^> 



d, , ^ sin « J , 
r ^ COS y d g* -|- ,^ dt. 



The constant logarithm to be used here is log^ y' I = 8.0850664436. 

21. 

In the HYPERBOLA, 9 and U would become imaginary quantities, to avoid 
which, other auxiliary quantities must be introduced in the place of them. We 
have already designated by ijJ the angle of which the cosine =-, and we have 
found the radius vector 



Sect. 1.] TO position in the orbit. 23 



2ecos^(v — ilj) cos^(v-{-xp)' 
For v=0, the factors cos i (v — xp), and cos ^ (v -\-xp), in the denominator of this 
fraction become equal, the second vanishes for the greatest positive value of v, 
and the first for the greatest negative value. Putting, therefore, 

cos ^(v — -w) 

cos I- (v + 'V^) ' 

we shall have m = 1 in perihelion ; it will increase to infinity as v approaches its 
limit 180° — If; on the other hand it will decrease indefinitely as v is supposed 
to return to its other limit — (180° — if) ; so that reciprocal values of ««, or, what 
amounts to the same thing, values whose logarithms are complementary, corre- 
spond to opposite values of v. 

This quotient u is very conveniently used in the hyperbola as an auxiliary 
quantity ; the angle, the tangent of which is 

can be made to render the same service with almost equal elegance ; and in order 
to preserve the analogy with the ellipse, we will denote this angle by ^ F. In 
this way the following relations between the quantities v, r, ti, F are easily brought 
together, in which we put a = — ^, so that h becomes a positive quantity. 

I. J=j»cotan^if 

n. r = ^ ^= pcostft 

l-j-ecosv 2cos^(v — \pi)(ios^{v-\--\p) 

m. tan^i?'=tan^yt/^ = tan^ytan|w = ^^ 

Y 1 = i Tm J- 1^ 1 -|- COS 1/> COS t> e -j- COS ^ 

cos^ ^ ~^ u' 2cos^(y — 1/;) cos ^ («;-[- If) l + ecosv* 

By subtracting 1 from both sides of equation V. we get, 



24 KELATIONS PERTAINING SIMPLY [BoOK I. 

In the same manner, by adding 1 to both sides, it becomes 
VII. cosiW = oosiJ'y/^j,p^^^=cosiJy/(fj=i)i 

By dividing VI. by VII. we should reproduce III. : the multiplication produces 
VIII. rsmv=p cotan if tan F=i tan f tan F 

= hp cotan i^ [u )=z ^ h tan t// [u ) . 



From the combination of the equations II. V. are easily derived 
X. r co^v =zh (e =,)-=^lh(2e — u ), 



22. 

By the differentiation of the formula IV. (regarding if as a constant quantity) 
we get 

— = i (tan h (y-j- V) — ^^^ ^ (^ — tf)jd2>:= ^ ^°^ de;; 
hence, 

u tan 1/) ' 

or by substituting for r the value taken from X. 

rrde;=: JJtam/^ \\ e{\ -\ ) Mm. 

Afterwards by integrating in such a manner that the integral may vanish at the 
perihelion, it becomes 

/rrdy^^Hani// {\e{u ) — \ogv)=^ht\l psjiX-^- \i)=:.'kt\2X^v;i sjlsj {\-\- \i). 

The logarithm here is the hyperbolic j if we wish to use the logarithm from 
Brigg's system, or in general from the system of which the modulus = \, and 



Sect. 1.] to position in the orbit. 25 

the mass fi (which we can assume to be indeterminable for a body moving in an 
hyperbola) is neglected, the equation assumes the following form : — 

XL ne'^^^^ — logu^^-^, 

or by introducing F, 

leianF—\ogtsin{4:6°-{-iF) = ~. 

Supposing Brigg's logarithms to be used, we have 

log I = 9.6377843113,, log 11-=^ 7.8733657527 ; 
but a little greater precision can be attained by the immediate application of the 
hyperbolic logarithms. The hyperbolic logarithms of the tangents are found in 
several collections of tables, in those, for example, which Schulze edited, and still 
more extensively in the Magnus Canon Triangulor. LogaritJimiciis of Benjamin Uksin, 
Cologne, 1624, in which they proceed, by tens of seconds. 

Finally, formula XI. shows that opposite values of t correspond to reciprocal 
values of u, or opposite values of F and v, on which account equal parts of the 
hyperbola, at equal distances from the perihelion on both sides, are described in 
equal times. 

23. 

If we should wish to make use of the auxiliary quantity u for finding the 
time from the true anomaly, its value is most conveniently determined by means 
of equation IV. ; afterwards, formula II. gives directly, without a new calculation, 
p by means of r, or r by means of p. Having found u, formula XL will give the 
quantity -^, which is analogous to the mean anomaly in the ellipse and will be 

denoted by N, from which will follow the elapsed time after the perihelion transit. 
Since the first term of JSf, that is ^ ^u" ^^^^^ ^J means of formula Vm. be 
made = /f^"" , the double computation of this quantity will answer for testing 
its accuracy, or, if preferred, N can be expressed without u, as follows : — 

XII. iV= Itmypsmv ^^^ cos\{v — ^p) 

2 cos i (f -f- If)) cos ^{v — Mj) ^ cos i (v + -U") ' 

4 



26 



RELATIONS PERTAINING SBIPLY 



[Book I. 



Example. — Let e= 1.2618820, or iy = 37° 35' 0'', v = 18° 51' 0", log r = 
0.0333585. Then the computation for ii,p, h, N, t, is as follows: — 



log cos J(y — 1//) . . 9.9941706) 

logcos-i(y + i/;) . . 9.9450577) 

logr 0.0333585 

log 2 6 0.4020488 



logj? . . , 
log cotan^ If 



0.3746356 

0.2274244 



loo; h 



0.6020600 
9.4312985 



log sin y 9.5093258 

logX 9.6377843 

Compl. log sin T/7 . . 0.2147309 ' 

8.7931395 

First term of iV^=r . 0.0621069 

logw= 0.0491129 

N= 0.0129940 

log^^ 7.8733658) 

flog 5 0.9030900) 



hence, 



)gu . . 


. 0.0491129 


^^ = 


1.1197289 


uu^= 


1.2537928 



The other calculation. 

log(?a( — 1) . . . 9.4044793 

Compl. logM . . . 9.9508871 

log ^ 9.6377843 

logle 9.7999888 

8.7931395 



logiV 8.1137429 

Difference .... 6.9702758 

logt 1.1434671 

t = 13.91448 



24 

If it has been decided to carry out the calculation with hyperbolic logarithms, 
it is best to employ the auxiliary quantity F, which will be determined by equa- 
tion III., and thence JV by XI. ; the semi-parameter will be computed from the 
radius vector, or inversely the latter from the former by formula YIII. ; the 
second part of JV can, if desired, be obtained in two ways, namely, by means of the 
formula hyp. log tan (45° -\- ^ F), and by this, hyp. log cos h {v — t//) — hyp. log 
cos I [v -\- T/'). Moreover it is apparent that here where ^ = 1 the quantity N 



Sect. 1.] 



TO POSITION IN THE ORBIT. 



27 



will come out greater in the ratio 1 : X, than if Brigg's logarithms were used. 
Our example treated according to this method is as follows : — 



lug uaii 2 1// .... 
log tan kv 


9.2201009 


log tan ^^ . . . . 


8.7519188 


loa; e 


0.1010188 


log tan ^ 


9.0543366 


e tan F — 


9.1553554 
0.14300638 


hyp. log tan (45° -]- i J') = 


= 0.11308666 


N— 


0.02991972 


loffJc 


8.2355814 ) 


flogJ 


0.9030900 i 



^ J^=3°13'58'a2 



C. hyp. log cos h{v — 1/') 
C. hyp. log cos i [v -\-i^): 



0.01342266 
0.12650930 



Difference 



0.11308664 



logiV^ 8.4759575 

Difference 7.3324914 

\ogt 1.1434661 

t= 13.91445 



25. 

For the solution of the inverse problem, that of determining the true anomaly 
and the radius vector from the time, the auxiliary quantity u or F must be first 
derived from iV=: IkV^t by means of equation XI. The solution of this tran- 
scendental equation will be performed by trial, and can be shortened by devices 
analogous to those we have described in article 11. But we suffer these to pass 
without further explanation ; for it does not seem worth while to elaborate as 
carefully the precepts for the hyperbolic motion, very rarely perhaps to be exhib- 
ited in celestial space, as for the elliptic motion, and besides, all cases that can 
possibly occur may be solved by another method to be given below. After- 
wards F or u will be found, thence v by formula III., and subsequently r will be 
determined either by II. or VIII. ; v and r are still more conveniently obtained 
by means of formulas VI. and VII. ; some one of the remaining formulas can be 
called into use at pleasure, for verifying the calculation. 



28 



KELATIONS PERTAINING SIMPLY 



[Book I. 



26. 

Example. — Eetaining for e and I the same values as in the preceding example, 
let t = 65.41236 : v and r are required. Using Briggs's logarithms we have 

log?! 1.8166598 

log^^J-l .... 6.9702758 

log i\^ 8.7859356, whence N= 0.06108514. Prom this it is 

seen that the equation iV=^etan^ — log tan (45° -|- ^ i^) is satisfied by 
F=:i 25°24'27".66, whence we have, by formula III, 

log tan ii^ . . . . 9.3530120 
log tan ^Y . . . . 9.5318179 

log tan J y .... 9.8211941, and thus ^ y = 33° 31' 29".89, and v — 

67° 2' 59';78. Hence, there follows, 

C.logco3i(. + ,) . 0.21374T6| ^^^_^^ ^^^^^^^^ 

C.logcos|(y — -li^) . 0.0145197) 

° log tan (45° 4-^^) . . . 0.1992280 

log f^ 9.9725868 ^ \ -r ^ ) 

loff /- . . . . . . 0.2008541. 



27. 
If equation IV. is differentiated, considering m, v, if, as variable at the same 
time, there results, 

d M sin 1/; d u -]- sin v^yp r tan tl; 



dy + 



di//. 



u 2 COS -^ (v — 1/;) cos -g- (w -}- If ) p '" ' j9 cos \p 

By differentiating in like manner equation XL, the relation between the 
differential variations of the quantities u, xp, N, becomes, 

= (|,(l_|_i.)_l)dw.- (" — 1)^-^ 
\ ^ ' uu' uf 

or 

diV 



diV" 



2 u cos^ 1/; 



di//, 



-v~ = 7- d i( + -. d w . 

/. OU ^ b cos I/; ' 



Sect. 1.] to position in the okbit. 29 

Hence, by eliminating d u by means of the preceding equation we obtain 
-T- =n dv-\- (14- -)t dw, 



dbbt&nilJ T TiT /5 I 5 \ sin V tan 1/; ^ 
v = —, — ^di\^ — {- + -) T^dw 

Xrr \r ^ p / cost/' ' 

= —, — ^diV^ — (14- ^)^— d^//. 

Xrr \ ' r/sini/j ' 

28. 
By differentiating equation X., all the quantities r, h, e, u, being regarded as 
variables, by substituting 

^ sin t/» T 

ae = — ^ aw, 

cos^xp ' ' 

and eliminating du with the help of the equation between dJY, du, dijj, given in 
the preceding article, there results. 



dr = Td^ 



bbe(uu — 1) 



b ' 2lur 



^^+2^{(^' + ^)«^^^-(^^-.^)'^^^|^^- 



The coefficient of diV is transformed, by means of equation YIII., into ,-^^^ : but 
the coefficient of d Tf , by substituting from equation IV., 

u (sin y — sine;) ^ sin (i// — v), - (sin 1/; -j- sin v) = sin {ip -{- v), 

is changed into 

h sin xp cos v pcosv ^ 

cos^i/; siaxp ' 

so that we have 

-, r J , , 5 sin w -, ,7- , » cos v , 

' I sin ip ' sm V ' 

So far, moreover, as JY is considered a function of b and t, we have 

dJY=^df-^^db, 

which value being substituted, we shall have d r, and also d y in the preceding 
article, expressed by means of d ^f, d ^, d if. Finally, we have here to repeat our 



30 RELATIONS PERTAINING SMPLY [BoOK 1. 

previous injunction, that, if the variations of the angles v and v^ are conceived to 
be expressed, not in parts of the radius, but in seconds, either all the tenns con- 
taining d y, d T//, must be divided bj 206264.8, or all the remaining terms must be 
multiplied by this number. 

29. 

Since the auxiliary quantities y, E, M, employed in the ellipse obtain 

imaginary values in the hyperbola, it will not be out of place to investigate their 

connection with the real quantities of which we have made use: we add therefore 

the principal relations, in which we denote by i the imaginary quantity y^ — 1. 

1 
sin (p = e = 

•* cos \p 

tan (45° — I 9) = ^^^ = 2y J^ = /tan i t/; 

tan 9) = 1 cotan (45° — Icp) — | tan (45° — 29)= r^ 

cos Cjp = i tan i\i 

9 = 90° -f i log (sin 9 + i cos 9) r= 90° — i log tan (45° + i i/^) 

tan ^ ^ = i taniF = li^izil) 

- — =-=r ^ cotan i U -I- i tan i U=^ — i cotan F, 

sin jQ ' 



or 



or 



n -J -n i(uu 1) 

2m 

jTil Til l+Qni 7?7 — 

sinJ'' 



2m 
cotan F=i cotan i F — ^ tan ^ F = — 



, rr . . T7 i(uu — 1) 

tan^=«sm^=^ — r-~- 

uu-\-l 

■n 1 MM + 1 

cos E = ^ := „ ' 

COS ^ 2 M 

iF= log (cos ^4" *'si^ ^) = 



or 



F = i log u = i log (45° -\- h F) 

° 2m a 

The logarithms in these formulas are hyperbolic. 



Sect. 1.] to position in the okbit. 31 



30. 

Since none of the numbers which we take out from logarithmic and trigo- 
nometrical tables admit of absolute precision, but are all to a certain extent 
approximate only, the results of all calculations performed by the aid of these 
numbers can only be approximately true. In most cases, indeed, the common 
tables, which are exact to the seventh place of decimals, that is, never deviate 
from the truth either in excess or defect beyond half of an unit in the seventh 
figure, furnish more than the requisite accuracy, so that the unavoidable errors 
are evidently of no consequence : nevertheless it may happen, that in special 
cases the effect of the errors of the tables is so augmented that we may be 
obliged to reject a method, otherwise the best, and substitute another in its place. 
Cases of this kind can occur in those computations which we have just explained; 
on which account, it will not be foreign to our purpose to introduce here some 
inquiries concerning the degree of precision allowed in these computations by 
the common tables. Although this is not the place for a thorough examination 
of this subject, which is of the greatest importance to the practical computer, yet 
we will conduct the investigation sufficiently far for our own object, from which 
point it may be further perfected and extended to other operations by any one 
requiring it. 

31. 

Any logarithm, sine, tangent, etc. whatever, (or, in general, any irrational 
quantity whatever taken from the tables,) is liable to an error which may amount 
to a half unit in the last figure : we will designate this limit of error by w, which 
therefore is in the common tables = 0.00000005. If now, the logarithm, etc., 
cannot be taken directly from the tables, but must be obtained by means of inter- 
polation, this error may be slightly increased from two causes. In the first place, it is 
usual to take for the proportional part, when (regarding the last figure as unity) it 
is not an integer, the next greatest or least integer ; and in this way, it is easily 
perceived, this error may be increased to just within twice its actual amount. But 



32 KELATIONS PERTALNING SDIPLY [BoOK I, 

we shall pay no attention to this augmentation of the error, since there is no 
objection to our affixing one more than another decimal figure to the propor- 
tional part, and it is verj evident that, if the proportional part is exact, the inter- 
polated logarithm is not liable to a greater error than the logarithms given 
directly in the tables, so far indeed as we are authorized to consider the changes 
in the latter as uniform. Thence arises another increase of the error, that this 
last assumption is not rigorously true ; but this also we pretermit, because the 
effect of the second and higher differences (especially where the superior tables 
computed by Taylor are used for trigonometrical functions) is evidently of no 
importance, and may readily be taken into account, if it should happen to turn 
out a little too great. In all cases, therefore, we will put the maximum unavoid- 
able error of the tables = w, assuming that the argument (that is, the number the 
logarithm of which, or the angle the sine etc. of which, is sought) is given with 
strict accuracy. But if the argument itself is only approximately known, and 
the variation w' of the logarithm, etc. (which may be defined by the method of 
differentials) is supposed to correspond to the greatest error to which it is liable, 
then the maximum error of the logarithm, computed by means of the tables, can 
amount to w -|- (a' . 

Inversely, if the argument corresponding to a given logarithm is computed 
by the help of the tables, the greatest error is equal to that change in the argu- 
ment which corresponds to the variation w in the logarithm, if the latter is cor- 
rectly given, or to that which corresponds to the variation w -f- w' in the loga- 
rithm, if the logarithm can be erroneous to the extent of w'. It will hardly be 
necessary to remark that w and 0/ must be affected by the same sign. 

If several quantities, correct within certain limits only, are added together, 
the greatest error of the sum will be equal to the sum of the greatest individual 
errors affected by the same sign ; wherefore, in the subtraction also of quantities 
approximately correct, the greatest error of the difference will be equal to the 
sum of the greatest individual errors. In the multiplication or division of a 
quantity not strictly correct, the maximum error is increased or diminished in the 
same ratio as the quantity itself 



Sect. 1.] to position in the orbit. 33 

32. 

Let us proceed now to the application of these principles to the most useful 
of the operations above explained. 

I. If cp and ^ are supposed to be exactly given in using the formula VII., 
article 8, for computing the true anomaly from the eccentric anomaly in the 
elliptic motion, then in log tan (45° — i (p) and log tan ^^, the error w maybe 
committed, and thus in the difference = log tan i v, the error 2 w ; therefore the 
greatest error in the determination of the angle i v will be 

ZmA\v 3 CO sin w 

d log tan \ V 2 X ' 

X denoting the modulus of the logarithms used in this calculation. The error, 
therefore, to which the true anomaly v is liable, expressed in seconds, becomes 

^^^' 206265 = 0".0712 sin v, 

if Bri^g's logarithms to seven places of decimals are employed, so that we may 
be assured of the value of y within 0".07 ; if smaller tables to five places only, are 
used, the error may amount to 7'M2. 

II. If e cos E is computed by means of logarithms, an error may be committed 
to the extent of 

3 CO e cos ^ 
I ' 
therefore the quantity 

1 — ecosK or - , 

will be liable to the same error. In computing, accordingly, the logarithm of this 
quantity, the error may amount to (1 -{-(?) w, denoting by d the quantity 

3 e cos J? 
1 — e cos ^ 

taken positively : the possible error in log r goes up to the same limit, log a being 
assumed to be correctly given. If the eccentricity is small, the quantity d is 
always confined within narrow limits; but when e differs but little from 1, 
1 — ecosE remains very small as long as E is small ; consequently, 8 may 

5 



34 RELATIONS PERTAINING SBIPLY [BoOK I. 

increase to an amount not to be neglected : for this reason formula III., article 8, 
is less suitable in this case. The quantity <^ may be expressed thus also, 

3 (a — r) 3 e (cos v-\-e) 

r 1 — ee ' 

which formula shows still more clearly when the error (1 -|-^) w may be neglected. 

III. In the use of formula X., article 8, for the computation of the true from 
the mean anomaly, the logu- is liable to the error {^ -{- ^d)o), and so the log 
sin ^(psmU u - to that of ( | -|- i^ <5') co ; hence the greatest possible error in the 
determination of the angles v — ^ or y is 

or expressed in seconds, if seven places of decimals are employed, 

(O'aee + 0".024 d) tan^v — U). 

When the eccentricity is not great, d and tan ^ {v — U) will be small quantities, 
on account of which, this method admits of greater accuracy than that which 
we have considered in I. : the latter, on the other hand, will be preferable 
when the eccentricity is very great and approaches nearly to unity, where d and 
tan i {v — U) may acquire very considerable values. It will always be easy to 
decide, by means of our formulas, which of the two methods is to be preferred. 

IV. In the determination of the mean anomaly from the eccentric by means 
of formula XII., article 8. the error of the quantity e sin U, computed by the help 
of logarithms, and therefore of the anomaly itself, M, may amount to 

3 cue sin ^ 
■ I ' 

which limit of error is to be multiplied by 206265" if wanted expressed in 
seconds. Hence it is readily inferred, that in the inverse problem where U is to 
be determined from 31 by trial, U may be erroneous by the quantity 

^J!llp^, ^^. 206265^^::= ^'" 7^'^" --. 206265", 

even if the equation E — esin^= 3/ should be satisfied with all the accuracy 
which the tables admit. 



Sect. 1.] 



TO POSITION IN THE ORBIT. 



35 



The true anomaly therefore computed from the mean may be incorrect in 
two ways, if we consider the mean as given accurately; first, on account of the 
error committed in the computation of v from E, which, as we have seen, is of 
slight importance ; second, because the value of the eccentric anomaly itself may 
be erroneous. The effect of the latter cause will be expressed by the product of 



the error committed in E into -r^, which product becomes 

^"7°^ . i^. 206265" = ^-^^^^. 206265'' = ('-^^^^^til^^^^")0''.0712, 

if seven places of decimals are used. This error, always small for small values of 
e, may become very large when e differs but little from unity, as is shown by the 
following table, which exhibits the maximum value of the preceding expression 
for certain values of e. 



e 


maximum error. 


e 


maximum error. 


e 


maximum eiTor. 


0.90 
0.91 
0.92 
0.93 


0".42 
0.48 
0.54 
0.62 


0.94 
0.95 
0.96 
0.97 


0".73 
0.89 
1 .12 
1 .50 


0.98 
0.99 
0.999 


2".28 

4.59 

46.23 



V. In the hyperbolic motion, if v is determined by means of formula III, 
article 21, from F and if accurately known, the error may amount to 

S_^pv^ 206265'^; 
but if it is computed by means of the formula 



tan i V ■■ 



(u — 1) tani-i/; 



M+1 ' 

u and tf being known precisely, the limit of the error will be one third greater, 
that is. 



4 o) sin V 



for seven places. 

VI. If the quantity 



. 206265" = 0':09 sin y 



— — iV" 



is computed by means of formula XI., article 22, with the aid of Briggs's loga- 



36 RELATIONS PERTAINING SIMPLY [BoOK I. 

rithms, assuming e and u or e and F to be known exactly, the first part will be 
liable to the error 

5 (mm — l)e(u 
2^^ » 



if it has been computed in the form 



or to the error 



if computed in the form 



Ae(M — 1)(m + 1)^ 
2« ^ 



3 (mm-|-1) ecu 



n^M-^; 



or to the error Sew tan F if computed in the form X e tan jP, provided we neglect 
the error committed in log X or log i h In the first case the error can be 
expressed also by 5 e w tan F, in the second by 7^, whence it is apparent that 
the error is the least of all in the third case, but will be greater in the first or 
second, according as ^< or - ^^ 2 or <^ 2, or according as +-^^ 36° 52' or <[ 36° 52'. 

But, in any case, the second part of N will be liable to the error w, 

VII. On the other hand, it is evident that if m or J' is derived from iV^ by 
trial, u would be liable to the error 

(w + 5 e to tan F) -r^, 

or to 

/ , Zeoi ^ Au 

according as the first term in the value of N is used separated into factors, or into 
terms ; F, however, is liable to the error 

(tu + 3 e w tani^) j-^ . 

The upper signs serve after perihelion, the lower before perihelion. Now if 
■^ is substituted here for -r^ or for -tjt, the effect of this error appears in 
the determination of v, which therefore will be 



Sect. 1.] 



TO POSITION IN THE ORBIT. 



37 



hhtamp (l±Se tan F) w hh tamp (1 -f- 3 e secF) co 
Irr Xrr ' 

if the auxiliary quantity u has been employed ; on the other hand, if F has been 
used, this effect becomes, 



J 5 tan 1/^ (1 + 3 e tan jP) CO a ( (1 -|- e cos vy , 

Irr X \ tan**!/; — 



3 e sin tJ (1 -|- e cos v) 
tan^tl; 



If the error is to be expressed in seconds, it is necessary to apply the factor 
206265". It is evident that this error can only be considerable when i/^ is a small 
angle, or e a little greater than 1. The following are the greatest values of this 
third expression, for certain values of e, if seven places of decimals are employed: 



e 


maximum error. 


1.3 


0".34 


1.2 


0.54 


1.1 


1.31 


1.05 


3 .03 


1.01 


34.41 


1.001 


1064 .65 



To this error arising from the erroneous value of F or u it is necessary to 
apply the error determined in Y. in" order to have the total uncertainty of v. 

VIII. If the equation XI., article 22, is solved by the use of hyperbolic loga- 
rithms, F being employed as an auxiliary quantity, the effect of the possible 
error in this operation in the determination of v, is found by similar reasoning 
to be, 

(1 -)- e cos w)^ w' I 3 e sin w (1 -j- e cos zj) ft) 
tan^ yp — I tan^ ■^j ' 

where by oi' we denote the greatest uncertainty in the tables of hyperbolic logar 
rithms. The second part of this expression is identical with the second part of 
the expression given in VIL; but the first part in the latter is less than the first 
in the former, in the ratio I to' : w, that is, in the ratio 1 : 23, if it be admissible 
to assume that the table of Ursin is everywhere exact to eight figures, or 

oi' == 0.000000005. 



38 KELATIONS PERTAINI2vG SBIPLY [BoOK I. 



33. 

The methods above treated, both for the determination of the true anomaly 
from the time and for the determination of the time from the true anomaly,* do 
not admit of all the precision that might be required in those conic sections of 
which the eccentricity differs but little from unity, that is, in ellipses and hyper- 
bolas which approach very near to the parabola; indeed, unavoidable errors, 
increasing as the orbit tends to resemble the parabola, may at length exceed all 
limits. Larger tables, constructed to more than seven figures would undoubtedly 
diminish this uncertainty, but they would not remove it, nor would they prevent 
its surpassing all limits as soon as the orbit approached too near the parabola. 
Moreover, the methods given above become in this case very troublesome, since a 
part of them require the use of indirect trials frequently repeated, of which 
the tediousness is even greater if we work with the larger tables. It certainly, 
therefore, will not be superfluous, to furnish a peculiar method by means of 
which the uncertainty in this case may be avoided, and sufficient precision may 
be obtained with the help of the common tables. 

34. 

The common method, by which it is usual to remedy these inconveniences, 
rests upon the following principles. In the ellipse or hyperbola of which e is the 
eccentricity, 'p the semi-parameter, and therefore the perihelion distance 

let the true anomaly v correspond to the time t after the perihelion; in the 
parabola of which the semi-parameter = 2 ^, or the perihelion distance = q, let 
the true anomaly w correspond to the same time, supposing in each case the 
mass \x, to be either neglected or equal. It is evident that we then have 



* Since the time contains the factor a^ or S^, the greater the values of a = , ^ - , or 5= ^ , 

" 1 — ee e^ — 1 

the more the error in illf or i^Twill be increased. 



Sect. 1.] to position in the orbit. 39 

J (l+ecosi;)2- j (1+cosm;)2 —\F'M^1^ 

the integrals commencing from ^ = and tv ^ 0, or 

J (l+ecosw)V2 J (l+cos«;)2 
Denoting y-^ by a, tan ^vhy&, the former integral is found to be 

y/(l_^a). (^^_^|^3(i_2«)_|^5(2a_3cca) + |^^(3«a — 4«3) — etc.), 

the latter, tan h w -\- Han^ ^ iv. From this equation it is easy to determine to 

by a and t', and also vhy a and ^^ by means of infinite series : instead of a may 

be introduced, if preferred, 

-1 2 « jt, 

1 — ez=i--T— ^=0. 

\-\-a 

Since evidently for « = 0, or (^ = 0, we have v^=-w, these series will have the 
following form : — 

w = v-\-dv' -^ddv" ^ dH'" + etc. 

where v', v", v'", etc. will be functions of v, and tv', tv\ iv" , functions of iv. When 
d is a very small quantity, these series converge rapidly, and few terms suffice for 
the determination of w from v, or of v from iv. i is derived from iv, or lo from t, 
by the method we have explained above for the parabolic motion. 

35. 

Our Bessel has developed the analytical expressions of the three first coeffi- 
cients of the second series iv', w" , iv'", and at the same time has added a table con- 
structed with a single argument lo for the numerical values of the two first lo' 
and iv", ( Von Zach Moimtliclie Correspondenz, vol. XIL, p. 197). A table for the 
first coefficient w', computed by Simpson, was already in existence, and was 
annexed to the work of the illustrious Olbers above commended. By the use 
of this method, with the help of Bessel's table, it is possible in most cases to 
determine the true anomaly from the time with sufficient precision; what remains 
to be desired is reduced to nearly the following particulars : — 



40 EELATIONS TERTAINIXG SQIPLY [BoOK I. 

I. In the inverse problem, the determination of the time, that is, from the 
true anomaly, it is requisite to have recourse to a somewhat indirect method, and 
to derive iv from v by trial. In order to meet this inconvenience, the first series 
should be treated in the same manner as the second : and since it may be readily 
perceived that — v' is the same function of v as to' of tv, so that the table for iv 
might answer for v' the sign only being changed, nothing more is required than 
a table for v", by which either problem may be solved with equal precision. 

Sometimes, undoubtedly, cases may occur, where the eccentricity differs but 
little from unity, such that the general methods above explained may not appear 
to afford sufficient precision, not enough at least, to allow the effect of the third 
and higher powers of d in the peculiar method just sketched out, to be safely 
neglected. Cases of this kind are possible in the hyperbolic motion especially, in 
which, whether the former methods are chosen or the latter one, an error of 
several seconds is inevitable, if the common tables, constructed to seven places of 
figures only, are employed. Although, in truth, such cases rarely occur in prac- 
tice, something might appear to be wanting if it were not possible in all cases to 
determine the true anomaly within 0".l, or at least 0".2, without consulting the 
larger tables, which would require a reference to books of the rarer sort. We 
hope, therefore, that it will not seem wholly superfluous to proceed to the exposi- 
tion of a peculiar method, which we have long had in use, and which will also 
commend itself on this account, that it is not limited to eccentricities differing but 
little from unity, but in this respect admits of general application. 

36. 

Before we proceed to explain this method, it will be proper to observe that 
the uncertainty of the general methods given above, in orbits approaching the 
form of the parabola, ceases of itself, when E or F increase to considerable mag- 
nitude, which indeed can take place only in large distances from the sun. To 
show which, we give to 

So^eosin.^ 206265'', 

the greatest possible error in the ellipse, which we find in article 32, IV., the 
following form, 



Sect. 1.] 



TO POSITION IN THE ORBIT. 



41 



<)e\l (1 — ee). s\nE 
l{\—ecosEY • 



206265'' 



from which is evident of itself that the error is always circumscribed within 
narrow limits when E acquires considerable value, or when cos E recedes further 
from unity, however great the eccentricity may be. This will appear still more 
distinctly from the following table, in which we have computed the greatest 
numerical value of that formula for certain given values of E, for seven decimal 
places. 



^=10° 


maximum error 


= rM 


20 




0.76 


30 




■ .34 


40 




.19 


50 




.12 


60 




.08 



The same thing takes place in the hyperbola, as is immediately apparent, if the 
expression obtained in article 32, VII., is put into this form, 

1) 



CO cos-F (cos^-|- 3 e sin F) \J (e 
' l(e — cosFy 



206265". 



The following table exhibits the greatest values of this expression for certain 
given values of E. 



F 


« 


maximum error. 


10° 


1.192 


0.839 


8".66 


20 


1.428 


0.700 


1 .38 


30 


1.732 


0.577 


0.47 


40 


2.144 


0.466 


0.22 


50 


2.747 


0.364 


0.11 


60 


3.732 


0.268 


0.06 


70 


5.671 


0.176 


0.02 



When, therefore, E or E exceeds 40° or 50° (which nevertheless does not easily 
occur in orbits differing but little from the parabola, because heavenly bodies 
moving in such orbits at such great distances from the sun are for the most part 
withdrawn from our sight) there will be no reason for forsaking the general 
method. For the rest, in such a case even the series which we treated in article 



42 EELATIONS PERTAINING SQITLY [BoOK I. 

34 might converge too slowly ; and therefore it is by no means to be regarded 
as a defect of the method about to be explained, that it is specially adapted 
to those cases in which E or F has not yet increased beyond moderate values. 

37. 

Let us resume in the elliptic motion the equation between the eccentric 
anomaly and the time, 

E — e sm ^ = - V V -rr/ 

where we suppose E to be expressed in parts of the radius. Henceforth, we 
shall leave out the factor Y^(l-|-i"'); if a case should occur where it is worth 
while to take it into account, the symbol t would not express the time itself after 
perihelion, but this time multiplied by y'(l -j-j"-)- We designate in future by q the 
perihelion distance, and in the place of j&and sin ^, we introduce the quantities 
^— sin^, and E— J^ [E— sin E) = J^ ^-j- J^ sin ^: 

the careful reader will readily perceive from what follows, our reason for selecting 
particularly these expressions. In this way our equation assumes the following 
form : — 

(l-.)(^V^+TVsin^) + (xV + TV^)(^-sm^) = ^^C-^)^ 
As long as E is regarded as a quantity of the first order, 

T'o^+TVsin^=^— eV^' + TiVo^' — etc. 
will be a quantity of the first order, while 

^— sin^=i^« — ^io^'+5oVo^' — etc., 
will be a quantity of the third order. Putting, therefore, 

4^=^^— 3V^'-WTo^'-etc. 
will be a quantity of the second order, and 

-^=1 + 2/0(7^'- etc. 
will differ from unity by a quantity of the fourth order. But hence our equation 
becomes 



Sect. 1.] 



TO POSITION IN THE ORBIT. 



^(2(l-.)^^-+^V(l-f9.)/) = ^"^(^)^ 



43 



[1] 



By means of the common trigonometrical tables, j\ E -f- iV ^i^ ^ "^^7 be com- 
puted with sufficient accuracy, but not E — sin ^ when ^ is a small angle; in this 
way therefore it would not be possible to determine correctly enough the quan- 
tities A and B. A remedy for this difficulty would be furnished by an appro- 
priate table, from which we could take out with the argument E, either B or the 
logarithm of B ; the means necessary to the construction of such a table will 
readily present themselves to any one even moderately versed in analysis. By 
the aid of the equation 

9^-f sin^ 



20 5 



U, 



^ A can be determined, and hence t by formula [1] with all desirable precision. 

The following is a specimen of such a table, which will show the slow increase 
of log B ; it would be superfluous to take the trouble to extend this table, for 
further on we are about to describe tables of a much more convenient form. 



E 


log 5 


E 


log 5 


E 


logB 


0° 


0.0000000 


25° 


0.0000168 


50° 


0.0002675 


5 


00 


30 


0349 


55 


3910 


10 


04 


35 


0645 


60 


5526 


15 


22 


40 


1099 






20 


69 


45 


1758 







38. 

It will not be useless to illustrate by an example what has been given in the 
preceding article. Let the proposed true anomaly =z 100°, the eccentricity 
= 0.96764567, log q = 9.7656500. The following is the calculation for E, B, 
A, and t : — 

log tan ^y 0.0761865 

logv/^ 9.1079927 

^ ~r ^ 

log tan J^ 9.1841792, whence I ^= 8°4ri9''.32, and ^ = 



44 RELATIONS PERTAINING SMPLY [BoOK I. 

17° 22' 38".64. To this value of E corresponds log B = 0.0000040 ; next is found 
in parts of the radius,^ = 0.3032928, sin E= 0.2986643, whence 2^ ^+ oV sin^ 
= 0.1514150, the logarithm of which = 9.1801689, and so log 4* = 9.1801649. 
Thence is derived, by means of formula [1] of the preceding article, 

'»g.T(?i- • • 2.4589614 ^o^-^^^J. . . 3.7601038 



1 



logA^ 9.1801649 log^^ 7.5404947 

log 43.56386= . . 1.6391263 log 19.98014= 1.3005985. 

19.98014 



63.54400 = !{. 
If the same example is treated according to the common method, e sin E in 
seconds is found = 59610".79 = 16°33'30".79, whence the mean anomaly = 

49'7''.85 = 2947'^85. And hence from 

log /?: (^-^')^ = 1.6664302 

is derived t = 63.54410. The difference, which is here only yo^o'o' P^^'^ ^^ ^ ^^7? 
might, by the errors concurring, easily come out three or four times greater. 
It is further evident, that with the help of such a table for log B even the inverse 
problem can be solved with all accuracy, E being determined by repeated trials, 
so that the value of t calculated from it may agree with the proposed value. 
But this operation would be very troublesome : on account of which, we will now 
show how an auxiliary table may be much more conveniently arranged, indefinite 
trials be altogether avoided, and the whole calculation reduced to a numerical 
operation in the highest degree neat and expeditious, which seems to leave 
nothing to be desired. 

39. 

It is obvious that almost one half the labor which those trials would require, 
could be saved, if there were a table so arranged that log B could be immedi- 
ately taken out with the argument A. Three operations would then remain ; 
the first indirect, namely, the determination of A so as to satisfy the equation 



Sect. 1.] to position in the oebit. 45 

[1], article 37 ; the second, the determination of U from A and B, which may be 
done directly, either by means of the equation 

or by this, 

sin^=2^(4* — 1^*); 
the third, the determination of v from B by means of equation YII., article 8. 
The first operation, we will bring to an easy calculation free from vague trials ; 
the second and third, we will really abridge into one, by inserting a new quantity 
C in our table by which means we shall have no need of B, and at the same 
time we shall obtain an elegant and convenient formula for the radius vector. 
Each of these subjects we will follow out in its proper order. 

First, we will change the form of equation [1] so that the Barkerian table 
may be used in the solution of it. For this purpose we will put 

A^ = tan ^wJ ^~Q^ , 
from which comes 

75tan^^ + 25tan|#=^^^^^^^^i±H=^^ 
denoting by a the constant 

75^va+ii)^ 

If therefore B should be known, w could be immediately taken from the Barkerian 
table containing the true anomaly to which answers the mean motion -^ ; A will 
be deduced from w by means of the formula 

A= ^ tan^ h w, 
denoting the constant 

Now, although B may be finally known from A by means of our auxiliary table, 
nevertheless it can be foreseen, owing to its dijBfering so little from unity, that if 
the divisor B were wholly neglected from the beginning, w and A would be 
affected with a slight error only. Therefore, we will first determine roughly w 
and A, putting B = l', with the approximate value of A, we will find B in our 



45 RELATIONS PERTAINING SIMPLY [BoOK I. 

auxiliary table, witli which we will repeat more exactly the same calculation ; 
most frequently, precisely the same value of B that had been found from the 
approximate value of A will correspond to the value of A thus corrected, so that a 
second repetition of the operation would be superfluous, those cases excepted in 
which the value of ^ may have been very considerable. 

Finally, it is hardly necessary to observe that, if the approximate value of B 
should in any other way whatever be known from the beginning, (which may 
always occur, when of several places to be computed, not very distant from each 
other, -some few are already obtained,) it is better to make use of this at once in 
the first approximation : in this manner the expert computer will very often not 
have occasion for even a single repetition. "We have arrived at this most rapid 
approximation from the fact that B differs from unity, only by a difference of the 
fourth order, and is multiplied by a very small numerical coefficient, which advan- 
tage, as will now be perceived, was secured by the introduction of the quantities 
E — sin E, j^Q E -j- -^^ sin E, in the place of E and sin E. 

40. 

Since, for the third operation, that is, the determination of the true anomaly, 
the angle E is not required, but the tan ^ E only, or rather the log tan h E, that 
operation could be conveniently joined with the second, provided our table sup- 
plied directly the logarithm of the quantity 

taxi\E 

which differs from unity by a quantity of the second order. We have preferred, 
however, to arrange our table in a somewhat different manner, by which, not- 
withstanding the small extension, we have obtained a much more convenient 
interpolation. By writing, for the sake of brevity, T instead of the tan^ h E, the 
value oi A, given in article 37, 

l5{E—smE) 

is easily changed to 

T—iT^-\-^T^—^ r*-f|f r« — etc. 



A = 



-T^ T+^^ T-'-^ T' + ^^ r^-etc' 



Sect. 1.] to position in the orbit. 4Y 

in which the law of progression is obvious. Hence is deduced, by the inversion 
of the series, 

^ = 1 - 1 ^ + if 5 ^^ + ^f ^ ^^ + ^Hlf 5 ^' + TsWiVi^ ^' + etc. 
Putting, therefore, 

C will be a quantity of the fourth order, which being included in our table, we 
can pass directly to v from A by means of the formula, 

, . llA-e I A rtan-|w 

te°*'' = \/T^\/ l-M+0 = V(i-M+g) ' 
denoting by y the constant 



/ 5 + 5e 



In this way we gain at the same time a very convenient computation for the 
radius vector. It becomes, in fact, (article 8, YL), 

gcosH^ _ q _ (l-f^-f<7)g 

cos^iw (l + rjcos^if (i-j-i.J+(7)cos2iv* 

41. 

Nothing now remains but to reduce the inverse problem also, that is, the 
determination of the time from the true anomaly, to a more expeditious form of 
computation : for this purpose we have added to our table a new column for T. 
T, therefore, will be computed first from v by means of the formula 

1 -\-e ' 

then A and log B are taken from our table with the argument T, or, (which is 
more accurate, and even more convenient also), G and log B, and hence A by 
the formula 

(1+0) r. 

finally t is derived from A and B by formula [1], article 37. If it is desired to 
call into use the Barkerian table here also, which however in this inveise problem 



48 RELATIONS PERTAINING SDIPLY [BoOK I. 

has less effect in facilitating the calculation, it is not necessary to pay any regard 
to A, but we have at once 

ta.n^w=z tan i v \J j^^f^, 
and hence the time t, by multiplying the mean motion corresponding to the true 
anomaly, tv. in the Barkerian table, by — . 

42. 

We have constructed with sufficient fulness a table, such as we have just 
described, and have added it to this work, (Table L). Only the first part pertains 
to the ellipse ; we will explain, further on, the other part, which includes the 
hyperbolic motion. The argument of the table, which is the quantity A, proceeds 
by single thousandths from to 0.300 ; the log B and C follow, which quantities 
it must be understood are given in ten millionths, or to seven places of decimals, 
the ciphers preceding the significant figures being suppressed ; lastly, the fourth 
column gives the quantity T computed first to five, then to six figures, which 
degree of accuracy is quite sufficient, since this column is only needed to get the 
values of log B and G corresponding to the argument T, whenever t is to be 
determined from v by the precept of the preceding article. As the inverse prob- 
lem which is much more frequently employed, that is, the determination of v and 
r from i, is solved altogether without the help of T, we have preferred the quan- 
tity A for the argument of our table rather than T, which would otherwise have 
been an almost equally suitable argument, and would even have facilitated a little 
the construction of the table. It will not be unnecessary to mention, that all the 
numbers of the table have been calculated from the beginning to ten places, and 
that, therefore, the seven places of figures which we give can be safely relied upon; 
but we cannot dwell here upon the analytical methods used for this work, by a 
full explanation of which we should be too much diverted from our j^lan. 
Finally, the extent of the table is abundantly sufficient for all cases in which it 
is advantageous to pursue the method just explained, since beyond the limit 
^ = 0.3, to which answers r= 0.392374, or ^=64° 7', we may, as has been 
shown before, conveniently dispense with artificial methods. 



Sect. 1.] 



TO POSITION EN THE ORBIT. 



49 



43. 

We add, for the better illustration of the preceding investigations, an example 
of the complete calculation for the true anomaly and radius vector from the time, 
for which purpose we will resume the numbers in article 38. We put then e = 
0.9674567, log ^=: 9.7656500, ^5=: 63.54400, whence, we first derive the constants 
log a = 0.03052357, log (i = 8.2217364, log y = 0.0028755. 

Hence we have log at= 2.1083102, to which corresponds in Barker's table 
the approximate value of w = 99° 6' whence is obtained A= 0.022926, and from 
our table log B = 0.0000040. Hence, the correct argument with which Barker's 
table must be entered, becomes log-^ = 2.1083062, to which answers to = 99° 6' 
13".14 ; after this, the subsequent calculation is as follows : — 



logtanH^; . . . 0.1385934 

log fi 8.2217364 

log J. 8.3603298 

A= 0.02292608 



log tan iw 0.0692967 

log 7 0.0028755 

i Comp. log(l— iA-{-C). 0.0040143 



log tan H' 0.0761865 

hence log B in the same manner as before ; i v= 50° 0' 0" 



0= 
iA^C= 



0.0000242 
0.9816833 
1.0046094 



log^ 

2 Comp. log cos I V 
log(l-|^+^. . 

c.iog(i4-i^4-c^. 



100 

9.7656500 
0.3838650 
9.9919714 
9.9980028 



logr 0.1394892 

If the factor B had been wholly neglected in this calculation, the true anomaly 
would have come out affected with a very slight error (in excess) of 0".l only. 



' 44. 

It will be in our power to despatch the hyperbolic motion the more briefly, 
because it is to be treated in a manner precisely analogous to that which we 
have thus far expounded for the elliptic motion. 

7 



50 RELATIONS PERTAINING SIMPLY [BoOK 1. 

We present the equation between the time t and the auxiliary quantity u in 
the following form : — 

in which the logarithms are hyperbolic, and 

2V(^-^) + Aiog^^ 

is a quantity of the first order, 

a quantity of the third order, when log u may be considered as a small quantity 
of the first order. Putting, therefore, 

A will be a quantity of the second order, but B will differ from unity by a differ- 
ence of the fourth order. Our equation will then assume the following form : — 

B{2{e-l)A^-^r^A'^-\-^e)J^)=M{^^f [2] 

which is entirely analogous to equation [1] of article 37. Putting moreover, 

T will be a quantity of the second order, and by the method of infinite series 
will be found 

Wherefore, putting 

C will be a quantity of the fourth order, and 

^— i_|2' • 
Finally, for the radius vector, there readily follows from equation VII., article 21, 

(1 — T)CQS'^V~{1 — ^A^0)C06^^V' 



Sect. 1.] to position in the orbit. 51 



45. 

The latter part of the table annexed to this work belongs, as we have remarked 
above, to the hyperbolic motion, and gives for the argument A (common to both 
parts of the table), the logarithm of B and the quantity to seven places of 
decimals, (the preceding ciphers being omitted), and the quantity T to five and 
afterwards to six figures. The latter part is extended in the same manner as 
the former to ^=0.300, corresponding to which is r= 0.241207, m= 2.930, 
or = 0.341, i^=+52°19'; to extend it further would have been superfluous, 
(article 36). 

The following is the arrangement of the calculation, not only for the determi- 
nation of the time from the true anomaly, but for the determination of the true 
anomaly from the time. In the former problem, T will be got by means of the 
formula 

^=^-^tanHe^; 

e-\- i 

with T our table will give log B and C, whence will follow 

finally t is then found from the formula [2] of the preceding article. In the last 
problem, will first be computed, the logarithms of the constants 

5^-5 
f^ l+9e 

' — V l+9e* 

A will then be determined from t exactly in the same manner as in the elliptic 
motion, so that in fact the true anomaly w may correspond in Barker's table to 
the mean motion -^,and that we may have 

A= ^ tan^ i w ; 
the approximate value of A will be of course first obtained, the factor B being 



52 EELATIONS PERTATNING SIMPLY [BoOK I. 

either neglected, or, if the means are at hand, being estimated ; our table will 
then furnish the approximate value of B, with which the work will be repeated ; 
the new value of B resulting in this manner will scarcely ever suffer sensible cor- 
rection, and thus a second repetition of the calculation will not be necessary. C 
will be taken from the table with the corrected value of A, which being done we 

shall have, 

, , ytanjw {I -\- ^ A -\- G)q 

^^^*^""V(l+^^+^)' ^— (l-i-4+0)cos^i^;• 

From this it is evident, that no difference can be perceived between the formulas 
for elliptic and hyperbolic motions, provided that we consider §, A, and T, in the 
hyperbolic motion as negative quantities. 

46. 

It will not be unprofitable to elucidate the hyperbolic motion also by some 
examples, for which purpose we will resume the numbers in articles 23, 26. 

I. The data are e = 1.2618820, log ^ = 0.0201657, y=18°5r0'': ^ is 
required. We have 

2 log tan I y . . . . 8.4402018 log 7^ 7.5038375 

log^ 9.0636357 l<^g(lH-^)- • • 0.0000002 

'+^ C.log(l — 1^) . 0.0011099 

1^^^ • ^-^OSS^^^ log^ 7.5049476 

T:=z 0.00319034 

log^= 0.0000001 

0= 0.0000005 

l^g^Tli • • • 2-2^^^^^^ ^^^'-^F^i-drS ■ ' • 2.8843582 

log^* 8.7524738 log^* 6.2574214 

log 13.77584 = . . 1.1391182 log 0.138605 .=: 9.1417796. 

0.13861 

13.91445 = ^;. 

II. e and g remaining as before, there is given i! = 65.41236; v and r are 
required. We find the logarithms of the constants, 



Sect. 1.] 



TO POSITION IN THE ORBIT. 



53 



log a = 9.9758345 
log /? = 9.0251649 
log y = 9.9807646. 

Next we have log azf = 1.7914943, whence by Barker's table the approxhuate 
value of e^=70°3r44", and hence ^ = 0.052983. To this A in our table 
answers log B = 0.0000207 ; from which, log ^ = 1.7914736, and the corrected 
value of w^ 70°3r36".86. The remaining operations of the calculation are as 
follows : — 

log tan ^ et- 9.8494699 

logy 9.9807646 

iC.log(l + |^4-C) . 9.9909602 



2 log tan Im; . . . 9.6989398 
log /? 9.0251649 



log^ 8.7241047 

A= . . . . . . 0.05297911 

log B as before, 

C= . . 0.0001252 

l^AA-\-C= . . 1.0425085 

1 — |iL+C= . . 0.9895294 



log tan iy 9.8211947 

iv=z ... 33°3r30''.02 
v= ... 67 3 .04 

log^ 0.0201657 

2 Clog cos ^y . . . . 0.1580378 
log (1 + 1^4- (7) . . 0.0180796 
C.log(l — ■1^4-C') . . 0.0045713 



logr 0.2008544 

Those which we found above (article 26), y = 67°2'69".78, logr = 0.2008541, 
are less exact, and v should properly have resulted = 67° 3' 0".00, with which 
assumed value, the value of ^ had been computed by means of the larger tables. 



SECOND SECTION. 



RELATIONS PERTAINING SIMPLY TO POSITION IN SPACE. 



47. 

In the first section, the motion of heavenly bodies in their orbits is treated 
without regard to the position of these orbits in space. For determining this 
position, by which the relation of the places of the heavenly body to any other 
point of space can be assigned, there is manifestly required, not only the position 
of the plane in which the orbit lies with reference to a certain known plane (as, 
for example, the plane of the orbit of the earth, the ecliptic), but also the position 
of the apsides in that plane. Since these things may be referred, most advanta- 
geously, to spherical trigonometry, we conceive a spherical surface described 
with an arbitrary radius, about the sun as a centre, on which any plane passing 
through the sun will mark a great circle, and any right line drawn from the 
sun, a point. For planes and right lines not passing through the sun, we draw 
through the sun parallel planes and right lines, and we conceive the great circles 
and points in the surface of the sphere corresponding to the latter to represent 
the former. The sphere may also be supposed to be described with a radius 
infinitely great, in which parallel planes, and also parallel right lines, are repre- 
sented in the same manner. 

Except, therefore, the plane of the orbit coincide with the plane of the ecliptic, 
the great circles corresponding to those planes (which we will simply call the orbit 
and the ecliptic) cut each other in two points, which are called nodes ; in one of 
these nodes, the body, seen from the sun, will pass from the southern, through the 
ecliptic, to the northern hemisphere, in the other, it will return from the latter to 
the former ; the former is called the ascending, the latter the descending node. We 
(54) 



Sect. 2.] TO position in space. 55 ' 

fix the positions of the nodes in the ecliptic by means of their distance from the 
mean vernal equinox [longitude) counted in the order of the signs. Let, in fig. 1, 
S2 be the ascending node, AQ> B part of the ecliptic, G Q, D part of the orbit ; 
let the motions of thg earth and of the heavenly body be in the directions from A 
towards B and from C towards D, it is evident that the spherical angle which 9> D 
makes with 9, B can increase from to 180°, but not beyond, without 9, ceasing 
to be the ascending node : this angle we call the inclination of the orbit to the 
ecliptic. The situation of the plane of the orbit being determined by the longi- 
tude of the node and the inclination of the orbit, nothing further is wanted 
except the distance of the perihelion from the ascending node, which we reckon 
in the direction of the motion, and therefore regard it as negative, or between 
180° and 360°, whenever the perihelion is south of the ecliptic. The following 
expressions are yet to be observed. The longitude of any j)oint whatever in 
the circle of the orbit is counted from that point which is distant just so far back 
from the ascending node in the orbit as the vernal equinox is back from the same 
point in the ecliptic : hence, the longitude of the perihelion will be the sum of the 
longitude of the node and the distance of the perihelion from the node ; also, the 
true longitude in oriit of the body will be the sum of the true anomaly and the 
longitude of the perihelion. Lastly, the sum of the mean anomaly and longitude 
of the perihelion is called the mean longitude : this last expression can evidently 
only occur in elliptic orbits. 

48. 

In order, therefore, to be able to assign the place of a heavenly body in space 
for any moment of time, the following things must be known. 

I. The mean longitude for any moment of time taken at will, which is called 
the epoch : sometimes the longitude itself is designated by the same name. For 
the most part, the beginning of some year is selected for the epoch, namely, noon 
of January 1 in the bissextile year, or noon of December 31 preceding, in the 
common year. 

IL The mean motion in a certain interval of time, for example, in one mean 
solar day, or in 365, 365^, or 36525 days. 



56 RELATIONS PERTAESriNG SIMPLY [BoOK I. 

III. The semi-axis major, whicli indeed might be omitted when the mass of 
the body is known or can be neglected, since it is already given by the mean 
motion, (article 7) ; both, nevertheless, are usually given for the sake of con- 
venience. 

IV. Eccentricity. V. Longitude of the perihelion. VI. Longitude of the 
ascending node. VII. Inclination of the orbit. 

These seven things are called the elements of the motion of the body. 

In the parabola and hyperbola, the time of passage through the perihelion 
serves in place of the first element ; instead of II., are given what in these 
species of conic sections are analogous to the mean daily motion, (see article 
19 ; in the hyperbolic motion the quantity X Jcb~^, article 23). In the hyperbola, 
the remaining elements may be retained the same, but in the parabola, where 
the major axis is infinite and the eccentricity = 1, the perihelion distance alone 
will be given in place of the elements III. and IV. 

49. 

According to the common mode of speaking, the inclination of the orbit, 
which we count from to 180°, is only extended to 90°, and if the angle made 
by the orbit with the arc Q, B exceeds a right angle, the angle of the orbit with 
the arc 9> A, which is its complement to 180°, is regarded as the inclination of 
the orbit ; in this case then it will be necessary to add that the motion is retrograde 
(as if, in our fiigure, E Q, F should represent a part of the orbit), in order that it 
may be distinguished from the other case where the motion is called direct. The 
longitude in orbit is then usually so reckoned that in Q, it may agree with the 
longitude of this point in the ecliptic, but decrease in the direction Q, F; the initial 
point, therefore, from which longitudes are counted contrary to the order of 
motion in the direction ^ F, is just so far distant from Q,, as the vernal equinox 
from the same Q, in the direction Q, A. Wherefore, in this case the longitude of 
the perihelion will be the longitude of the node diminished by the distance of 
the perihelion from the node. In this way either form of expression is easily con- 
verted into the other, but we have preferred our own, for the reason that we 
might do away with the distinction between the direct and retrograde motion, 



Sect. 2.] to position in space. 57 

and use always the same formulas for both, while the common form may fre- 
quently require double precepts. 

50. 

The most simple method of determining the position, with respect to the 
ecliptic, of any point whatever on the surface of the celestial sphere, is by means 
of its distance from the ecliptic {latitude), and the distance from the equinox of 
the point at which the ecliptic is cut by a perpendicular let fall upon it, {longi- 
tude). The latitude, counted both ways from the ecliptic up to 90°, is regarded as 
positive in the northern hemisphere, and as negative in the southern. Let the 
longitude X, and the latitude /5, correspond to the heliocentric place of a celestial 
body, that is, to the projection upon the celestial sphere of a right line drawn 
from the sun to the body ; let, also, u be the distance of the heliocentric place 
from the ascending node (which is called the argument of the latitude), i be the 
inclination of the orbit, S2 the longitude of the ascending node; there will exist 
between i, u, (i, X — Q, which quantities will be parts of a right-angled spherical 
triangle, the following relations, which, it is easily shown, hold good without any 
restriction : — 

I. tan (X — 9,) = cos ^ tan u 

11. tan fi = tan^ sin (X — Q) 

III. sin /i = sin i sin u 

IV. cos u = cos (3 cos (X — Q). 

When the quantities i and u are given, X — Q will be determined from them by 
means of equation I, and afterwards /5 by II. or by III., if /? does not approach 
too near to + 90° ; formula IV. can be used at pleasure for confirming the cal- 
culation. Formulas I. and IV. show, moreover, that X — Q and u always lie in 
the same quadrant when ^ is between 0° and 90° ; X — Q, and 360° — u, on the 
other hand, will belong to the same quadrant when ^ is between 90° and 180°, or, 
according to the common usage, when the motion is retrograde : hence the ambi- 
guity which remains in the determination of A, — Q by means of the tangent 
according to formula I., is readily removed. 



58 RELATIONS PERTAIMNG SEVITLT • [BoOK I. 

The following formulas are easily deduced from the combination of the pre- 
ceding : — 

V. sm{u — X -}- S ) = 2 sin^ I i sin it cos {I — Q ) 
VI. sin (t( — 2. -|- a ) = tan i i sin /? cos {X — Q) 
VII. sin (it — X -|- S^ ) = tan i i tan /:? cos u 
VIII. sin (tf + X — ^ ) = 2 cos^ I i sin u cos {I — 9,) 
IX. sin (m -|- >. — S2 ) = cotan h i sin /? cos (^ — 9,) 
X. sin (it -|- X — 0,)^= cotan ^ i tan /? cos m. 
The angle u — 1^9>, v;hen i is less than 90°, or u-\-l — 9, when i is more 
than 90°, called, according to common usage, the reduction to the ecliptic, is, in fact, 
the difference between the heliocentric longitude I and the longitude in orbit, 
which last is by the former usage g^ + w, by ours Q> -j- u. When the inclination 
is small or differs but little from 180°, the same reduction may be regarded as a 
quantity of the second order, and in this case it will be better to compute first ^ 
by the formula III., and afterwards I by VII. or X., by which means a greater 
precision will be attained than by formula I. 

If a perpendicular is let fall from the place of the heavenly body in space 
upon the plane of the ecliptic, the distance of the point of intersection from the 
sun is called the curtate distance. Designating this by /, the radius vector likewise 
by r, we shall have 

XL / = rcos|?. 

51. 

As an example, we will continue further the calculations commenced in arti- 
cles 13 and 14, the numbers of which the planet Juno furnished. We had 
found above, the true anomaly 315°r23".02, the logarithm of the radius vector 
0.3259877: now let « = 13°6'44".10, the distance of the perihelion from the 
node =241°10'20".57, and consequently « = 196°ir43".59 ; finally let 9 = 
171° 7'48".73. Hence we have : — 

logtauM .... 9.4630573 log sin (X— g2) . • . . 9.4348691n 

log cose .... 9.9885266 log tan e 9.3672305 

loglan {1—9). . 9.4515839 log tan /S 8.8020996 « 



Sect. 2.] 
I— S 



TO POSITION IN SPACE. 



59 



Q, = 195°4r40".25 

1= 6 55 28.98 

logr ..... . 0.3259877 

log cos /5 9.9991289 

log / 0.3251166 log cos 



/?= — 3°3r40':02 

log cos ^ 9.9991289 

loffcosX— a ... 9.9832852 » 



9.9824141% 
9.9824141%. 



The calculation by means of formulas III, VII. would be as foUows : — 



log sin ?{ . . ., . 9.4454714% 
log sin « 9.3557570 

log sin /5 .... 8.8012284% 
/?= — 3°3r40".02 



log tan i 2 9.0604259 

log tan ^ 8.8020995% 

log cos M 9.9824141% 



log 


sin (u — 


X-i-Q) 


. 7.8449395 


u — 


-l-\-Q 


= 


0°24' 8".34 




X—Q 


= 


195 47 40 .25. 



52. 

Regarding ^ and u as variable quantities, the differentiation of equation III., 
article 50, gives 

cotan /3 d/5 = cotan ^d^' -|- cotan udu, 
or 

XII. d|3 = sin(^ — ^)d^-|- sin ^ cos (X — 9,)du. 

In the same manner, by differentiation of equation I. we get 

XIII. d(X— Q) = — tiin(Uos{l— Q)di-{-^du. 
Finally, from the differentiation of equation XL comes 

dr = cos /? dr — r sin/3d/5, 



XIV. dr = cosjSdr — r sin ^ sin(X — Q)di — rsm^ sin ^ cos (I — Q)du. 

In this last equation, either the parts that contain di and du are to be divided by 
206265", or the remaining ones are to be multiplied by this number, if the 
changes of ^' and u are supposed to be expressed in minutes and seconds. 



60 RELATIONS PERTAINESIG SBIPLY [BoOK I. 

53. 

The position of any point whatever in space is most conveniently deter- 
mined by means of its distances from three planes cutting each other at right 
angles. Assuming the plane of the ecliptic to be one of these planes, and denot- 
ing the distance of the heavenly body from this plane by 2", taken positively on 
the north side, negatively on the south, we shall evidently have s = /tan /:J =: 
r sin /5 ^ r sin i sin u. The two remaining planes, which we also shall consider 
drawn through the sun, will project great circles upon the celestial sphere, which 
will cut the ecliptic at right angles, and the poles of which, therefore, will lie in 
the ecliptic, and will be at the distance of 90° from each other. We call that pole 
of each plane, lying on the side from which the positive distances are counted, 
the positive pole. Let, accordingly, iV and N -\- 90° be the longitudes of the 
positive poles, and let distances from the planes to which they respectively 
belong be denoted by x and i/. Then it will be readily perceived that we have 

a; = r'cos {X — N) 

= r cos {^} cos (A, — Q,) cos [N — 9,)-\-r cos ^i sin (X — ^) sin {N — Q, ) 
y = / sin (A, — iV) 
= r cos fi sin (^ — 9>) cos (iV — 9>) — r cos (i cos {I — ^) sin (iV — Q ), 
which values are transformed into 

x = r cos {jV — Q, ) cos u -|- r cos i sin (iV — Q, ) sin u 
2/ = r cos ^' cos {JV — 9,) sin u — rsin (iV — Q, ) cosu. 

If now the positive pole of the plane of x is placed in the ascending node, so that 
iV^= S2, we shall have the most simple expressions of the coordinates x,f/, z, — 

a; = r cos u 

y = r cos i sin u 

s = r sin i sin u . 

But, if this supposed condition does not occur, the formulas given above will 
still acquire a form almost equally convenient, by the introduction of four 
auxiliar}'^ quantities, a, h, A, B, so determined as to have 



Sect. 2,] TO POSITION m space. 61 

cos {JV — 9,) = a sin A 
cos i sin (iV — S^ ) = a cos A 
— sin {J\^— 9,) = bsmB 
cos i cos (iV — Q)=:ib cos B, 
(see article 14, II.). We shall then evidently have 
X =^ra sm{u -\- A) 
2/ = ri sin (u-\- B) 
g = r sin i sin u . 

54. 

The relations of the motion to the ecliptic explained in the preceding article, 
will evidently hold equally good, even if some other plane should be substituted 
for the ecliptic, provided, only, the position of the plane of the orbit in respect 
to this plane be known ; but in this case the expressions longitude and latitude 
must be suppressed. The problem, therefore, presents" itself: From the known 
position of the plane of the orbit and of another new plane in respect to the ecliptic, to 
derive the position of the plane of the orbit in respect to the neiv plane. Let nQ>, 9>9>', 
nO,' he parts of the great circles which the plane of the ecliptic, the plane of the 
orbit, and the new plane, project upon the celestial sphere, (fig. 2). In order 
that it may be possible to assign, without ambiguity, the inclination of the second 
circle to the third, and the place of the ascending node, one direction or the other 
must be chosen in the third circle, analogous, as it were, to that in the ecliptic 
which is in the order of the signs; let this direction in our figure be from n toward 
9>'. Moreover, of the two hemispheres, separated by the circle nO,', it will be 
necessary to regard one as analogous to the northern hemisphere, the other to 
the southern ; these hemispheres, in fact, are already distinct in themselves, since 
that is always regarded as the northern, which is on the right hand to one moving 
forward* in the circle according to the order of the signs. In our figure, then, Q,, 
n, Q,', are the ascending nodes of the second circle upon the first, the third upon 
the first, the second upon the third; 180° — nQQ>', Q, n Q,' , n Q>' Q, the inclina- 

* In the inner surface, ttat is to say, of the sphere represented by our figure. 



62 RELATIONS PERTAINING SEVIPLY [BouR 1. 

tions of the second to the first, the third to the first, the second to the third. 
Our problem, therefore, depends upon the solution of a spherical triangle, in 
which, from one side and the adjacent angles, the other parts are to be deduced. 
We omit, as sufficiently well known, the common precepts for this case given 
in spherical trigonometry : another method, derived from certain equations, which 
are sought in vain in our works on trigonometry, is more conveniently emploj'ed. 
The following are these equations, which we shall make frequent use of in future: 
a, h, c, denote the sides of the spherical triangle, and A, B, C, the angles oppo- 
site to them respectively : — 



I. 



&m\{b — c) sin ^ {B— G) 

sin -I a cos ^ A 



TT ?i? 2" (^ ~l~ ^) ^°^ 2 (-^ — ^) 



-,jy COS ^{h — c) &m^{B-\-G) 

coi\a cos|^^ 

j-y- cos 1(5 + c) cos \ {B-\- O) 



Although it is necessary, for the sake of brevity, to omit here the demonstration 
of these propositions, any one can easily verify them in Iriangles of which neither 
the sides nor the angles exceed 180°. But if the idea of the spherical triangle is 
conceived in its greatest generality, so that neither the sides nor the angles are 
confined within any limits whatever (which affords several remarkable advan- 
tages, but requires certain preliminary explanations), cases may exist in which it 
is necessary to change the signs in all the preceding equations ; since the former 
signs are evidently restored as soon as one of the angles or one of the sides is 
increased or diminished 360°, it will always be safe to retain the signs as we 
have given them, whether the remaining parts are to be determined from a side 
and the adjacent angles, or from an angle and the adjacent sides ; for, either 
the values of the quantities sought, or those differing by 360° from the true val- 
ues, and, therefore, equivalent to them, will be obtained by our formulas. We 
reserve for another occasion a fuller elucidation of this subject : because, in the 
meantime, it will not be difficult, by a rigorous induction, that is, by a complete 
enumeration of all the cases, to prove, that the precepts which we shall base upon 



Sect. 2.] TO position in space. 63 

these formulas, both for the solution of our present problem, and for other pur- 
poses, hold good in all cases generally. 

55. 

Designating as above, the longitude of the ascending node of the orbit upon 
the ecliptic by Q, the inclination by ^'; also, the longitude of the ascending node 
of the new plane upon the ecliptic by n, the inclination by e ; the distance of the 
ascending node of the orbit upon the new plane from the ascending node of the 
new plane upon the ecliptic (the arc n^' in fig. 2) by 9,', the inclination of the 
orbit to the new plane by i' ; finally, the arc from 9, to ^' in the direction of the 
motion by J : the sides of our spherical triangle will be Q, — n, Q,', J, and the 
opposite angles,/, 180° — /, e. Hence, according to the formulas of the preceding 
article, we shall have 

sin ^ i' sin ^ (9,' -\- J) =: sin i [Q — n) sin J («' -|- e) 
sin i i'cos ^ [9' -\- J) = cos i (9, — n) sin i [i — e) 
cos hi' sin h{9' — ^) = sin k {9 — oi) cos h (^' -|- e) 
cos|^''cos t {9' — z/) =cos ^ (S2 — n)co8h {i — £). 

The two first equations will furnish h [9' -\- J) and sin k i' ; the remaining two, 
h{9' — z/) and cos ^^''; from ^ {9' -\- J) and h {9' — z/) will follow S^'and//; 
from sin i ^' and cos ^ i' (the agreement of which will serve to prove the calcula- 
tion) will result i'. The uncertainty, whether h{9' -{- /I) and i {9' — /I) should 
be taken between and 180° or between 180° and 360°, will be removed in this 
manner, that both sin i i\ cos \ i\ are positive, since, from the nature of the case, i' 
must fall below 180°. 

56. 

It will not prove unprofitable to illustrate the preceding precepts by an 
example. Let g2 == 172°28'13".7, e= 34° 38' I'M; let also the new plane be 
parallel to the equator, so that n = 180° ; we put the angle e, which will be the 
obliquity of the ecliptic = 23° 27' 55".8. We have, therefore. 



64 



RELATIONS PERTAINING SIMPLY 



[Book 1. 



U 71 = 

^ -|- £ := 

log sin i (S2 — n) . . 
log sin i (/-)- e) . . . 
log cos 2 («' -f- ^) • • • 
, .; c V, ■ have 
log sin i i' sin k {9>' -\- J) 
log sin h I cos -^ { S ' -j- '^) 



-7°3r46".3 
58 5 56 .9 
11 10 5 .3 

. 8.8173026 ?z 
. 9.6862484 
. 9.9416108 



8.5035510« 

8.9872023 



^(Q_j?)= _3' 

H« + ^)= 29 

|(^ — e)= 5 

log cos ^ ( S? — n) . . 

logsin|(^' — e) . . . 

log cos \{i — £) . . . 

log cos h i' sin I ( S^ ' — /t) 
log cos hi' COS h {Q> ' — //) 



45'53".15 

2 58 .45 
35 2.65 
9.9990618 
8.9881405 
9.9979342. 

8.7589134^2 
9.9969960 



whence I ( 9,'-\- A) = 341° 49' 19".01 whence h{9.' — A) = 356° 41' 31".43 
log sin ^e" 9.0094368 log cos ^/ 9.9977202. 

Thus we obtain ^ ^•' = 5° 51' 56".445, ^" = 11° 43'52".89, S^'= 338° 30' 50".43, 
A = — 14° 52' 12".42. Finally, the point ?? evidently corresponds in the celestial 
sphere to the autumnal equinox ; for which reason, the distance of the ascending 
node of the orbit on the equator from the vernal equinox (its ri(/ht ascetision) 
will be 158°30'50".43. 

In order to illustrate article 53, we will continue this example still further, 
and will develop the formulas for the coordinates with reference to the three 
planes passing through the sun, of which, let one be parallel to the equator, and 
let the positive poles of the two others be situated in right ascension 0° and 90° : 
let the distances from these planes be respectively z, x, y. If now, moreover, 
the distances of the heliocentric place in the celestial sphere from the points 9,, 
S ', are denoted respectively by «, li, we shall have ?«'= « — A=iu-\- 14° 52' 12".42, 
and the quantities which in article 53 were represented by ^, N — 9> , ^<, will here 
be /, 180° — S^', iL Thus, from the formulas there given, follow, 

log a sin .4 . . . . 9.9687197?^ log^sin^ . . . . 9.5638058 
log a cos ^ . . . . 9.5546380;? log^cos^ . . . . 9.9595519 ?z 



whence ^ = 248°55'22".97 

log a 9.9987923 

We have therefore, 



whence^ = 158° 5' 54".97 

log* 9.9920848. 



Sect. 2.] to position in space. 65 

X = ar sin {i/-{-24:S° 6b' 22". 97) = ar sin (« + 263°47'3.5".39) 
t/ = br sin (^f' -[- 158 5 54 .97) = br sin {u + 172 58 7 .39) 
2r = crsinz/ = cr sin (^J -[- 14 52 12.42) 

in which log c = log sin i' = 9.3081870. 

Another solution of the problem here treated is found in Vo7i Zach's MonatUche 
Correspondens, B. IX. p. 385. 

57. 

Accordingly, the distance of- a heavenly body from any plane passing through 
the sun can be reduced to the form krmi [v -|- K), v denoting the true anomaly; 
Jc will be the sine of the inclination of the orbit to this plane, K the distance 
of the perihelion from the ascending node of the orbit in the same plane. So far 
as the position of the plane of the orbit, and of the line of apsides in it, and also 
the position of the plane to which the distances are referred, can be regarded as 
constant, Jc and K will also be constant. In such a case, however, that method 
will be more frequently called into use in which the third assumption, at least, is 
not allowed, even if the perturbations should be neglected, which always affect 
the first and second to a certain extent. This happens as often as the distances 
are referred to the equator, or to a plane cutting the equator at a right angle 
in given right ascension: for since the position of the equator is variable, owing to 
the precession of the equinoxes and moreover to the nutation (if the true and not 
the mean position should be in question), in this case also k and K will be subject 
to changes, though undoubtedly slow. The computation of these changes can be 
nlade by means of differential formulas obtained without difficulty: but here 
it may be, for the sake of brevity, sufficient to add the differential variations 
oii', Q,' and J, so far as they depend upon the changes of Q, — n and £. 

d/ = sin £ sin Q,' A [Q, — n) — cos S^'de 

T -^ , sin i cos A 



d(S^— n)-f- 



tarn 
sin ' 



^^^ Mn.cu.,6 d(S^— n)4---^iL^d£. 
sint ^ '^ ' sln^ 

Finally, when the problem only is, that several places of a celestial body with 

9 



66 RELATIONS PERTAKnNG SIMPLY [BoOK 1. 

respect to such variable planes may be computed, which places embrace a mod- 
erate interval of time (one year, for example), it will generally be most con- 
venient to calculate the quantities «, A, b, B, c, C, for the two epochs between 
which they fall, and to derive from them by simple interpolation the changes for 
the particular times proposed. 

58. 

Our formulas for distances from given planes involve v andr; when it is 
necessary to determine these quantities first from the time, it will be possible to 
abridge part of the operations still more, and thus greatly to lighten the labor. 
These distances can be immediately derived, by means of a very simple formula, 
from the eccentric anomaly in the ellipse, or from the auxiliary quantity F or ?f 
in the hyperbola, so that there will be no need of the computation of the true 
anomaly and radius vector. The expression kr sin {v -\- K) is changed ; 

I. For the ellipse, the symbols in article 8 being retained, into 

a^cosg) cos ^ sin ^-|- a^sin^(cos^ — e). 

Determining, therefore, I, L, X, by means of the equations 

aJcsm K^=lsinL 
ak cos(p cosK=lcosL 
— eak smK= — el smL=zX, 

our expression passes into Ism [U -\- Z) -{- X, in which /, X, X will be constant, so 
far as it is admissible to regard /c, K, e as constant ; but if not, the same precepts 
which we laid down in the preceding article will be sufficient for computing thei) 
changes. 

We add, for the sake of an example, the transformation of the expression for 
X found in article 56, in which we put the longitude of the perihelion = 121° 17' 
34".4, 9) = 14" 13'3r.97, log a = 0.4423790. The distance of the perihelion from 
the ascending node in the ecliptic, therefore, = 308° 49' 20".7 = ^( — v; hence 
K= 212°36'56".09. Thus we have. 



Sect. 2.] TO position in space. 67 

logaJc ..... 0.4411713 log/sinZ .... 0.1727000^2 
logsinX .... 9.7315887^^ log/cosX .... 0.3531154;? 
log«7fcos9 . . . 0.4276456 whence L = 213°25'5r.30 

log cos ^ .... 9.9254698 ?2. log/=: 0.4316627 

log;ir= 9.5632352 

1= +0.3657929. 

n. In the hyperbola the formula ^r sin (y -|-^), by article 21, passes into 
X -}- |U. tan F-\-v sec F, if we put eb k sin K= I, ik tan i/^ cos ir= ^i, — bk sinK 
=:v ; it is also, evidently, allowable to bring the same expression under the form 

cosF 
If the auxiliary quantity u is used in the place of F, the expression ^-r sin (v-\-S) 
will pass, by article 21, into 

in which a, (i, y, are determined by means of the formulas 

a = 1 = ehk sin K 

(j = ^v -\- fi,) =— i ehk sin {K—xi)) 

y ^ ^ {y — /^) = — h ebk sm. {^-\- <f )• 
III. In the parabola, where the true anomaly is derived directly from the time, 
nothing would remain but to substitute for the radius vector its value. Thus, 
denoting the perihelion distance by q, the expression A^r sin {v -\- K) becomes 



59. 

The precepts for determining distances from planes passing through the sun 
may, it is evident, be applied to distances from the earth ; here, indeed, only the 
most simple cases usually occur. Let R be the distance of the earth from the sun, 
L the heliocentric longitude of the earth (which differs 180° from the geocentric 
longitude of the sun), lastly, X, Y, Z, the distances of the earth from three planes 
cuttino; each other in the sun at riw:ht angles. Now if 



68 



RELATIONS PERTAINING SEMPLY 



[Book I. 



I. The plane of Z is the ecliptic itself, and the longitudes of the poles of the 
remaining planes, the distances from which are X, F, are respectively N, and 
i\r^ 90°; then 

X^i?cos(Z — i\^), r = i?sin(X — i\^), ^=0. 

n. If the plane of Z is parallel to the equator, and the right ascensions of the 
poles of the remaining planes, from which the distances are X., Y, are respectively 
0° and 90°, we shall have, denoting by £ the obliquity of the ecliptic, 

X=BgosL, Y :^ H cos e sin L, Z= E sine sin L. 

The editors of the most recent solar tables, the illustrious Von Zach and de 
Lambke, first began to take account of the latitude of the sun, which, produced 
by the perturbations of the other planets and of the moon, can scarcely amount 
to one second. Denoting by B the hehocentric latitude of the earth, which will 
always be equal to the latitude of the sun but affected with the opposite sign, we 
shall have, 



In Case I. 

X=BgosBcos{L — JY) 
Y=EcosBsin{L—JY) 
Z=E sin B 



In Case 11. 

X^= E cos B cos L 

Y=z E cos B cos e sin X — E sin B sin £ 

Z = E cos B sin esinL-\- E sin B cos e. 



It will always be safe to substitute 1 for cos B, and the angle expressed in parts 
of the radius for sin B. 

The coordinates thus found are referred to the centre of the earth. If ^, -)], C, 
are the distances of any point whatever on the surface of the earth from three 
planes drawn through the centre of the earth, parallel to those which were drawn 
through the sun, the distances of tliis point from the planes passing through the 
sun, will evidently be X-|- ^; ^-\-V^ Z -\-L: the values of the coordinates I, i], L, 
are easily determined in both cases by the following method. Let () be the radius 
of the terrestrial globe, (or the sine of the mean horizontal parallax of the sun,) 
I the longitude of the point at which the right line drawn from the centre of the 
earth to the point on the surface meets the celestial sphere, ('i the latitude of the 
same point, a the right ascension, d the declination, and we shall have, 



Sect. 2.] to position m space. 69 



In Case I. 
^ = QCOS^ COS (X JV) 

?j = ^ COS jS sin (^ — JV) 
C = 9 sin jS 



In Case II. 

^ = Q cosd COS a 
rj = Q cos d sia a 



This point of the celestial sphere evidently corresponds to the zenith of the 
place on the surface (if the earth is regarded as a sphere), wherefore, its right 
ascension agrees with the right ascension of the mid-heaven, or with the sidereal 
time converted into degrees, and its declination with the elevation of the pole ; 
if it should be worth while to take account of the spheroidal figure of the earth, 
it would be necessary to adopt for d the connected elevation of the pole, and for 
Q the true distance of the place from the centre of the earth, which are deduced 
by means of known rules. The longitude and latitude I and /3 will be derived 
from a and d by known rules, also to be given below : it is evident that I coin 
cides with the longitude of the nonagesimal, and 90° — 13 with its altitude. 

60. 

If X, y, z, denote the distances of a heavenly body from three planes cutting 
each other at right angles at the sun; X, Y, Z, the distances of the earth (either 
of the centre or a point on the surface), it is apparent that x — X,i/ — Y, z — Z^ 
would be the distances of the heavenly body from three planes drawn through 
the earth parallel to the former; and these distances would have the same relation 
to the distance of the body from the earth and its geocentric place^ (that is, the place 
of its projection in the celestial sphere, by a right hne drawn to it from the earth), 
which x,t/, gjhave to its distance from the sun and the heUocentric place. Let J 
be the distance of the celestial body from the earth ; suppose a perpendicular in 
the celestial sphere let fall from the geocentric place on the great circle which 
corresponds to the plane of the distances z, and let a be the distance of the 
intersection from the positive pole of the great circle which corresponds to the 



* In the broader sense : for properly this expression refers to that case in which the right line is 
drawn from the centre of the earth. 



70 RELATIONS PERTAINING SIMPLY [BoOK 1. 

plane of the distances x; and, finally, let h be the length of this perpendicular, or 
the distance of the geocentric place from the great chcle corresponding to the 
distances s. Then h will be the geocentric latitude or declination, according as the 
plane of the distances s is the ecUptic or the equator ; on the other hand, a -j- JV 
will be the geocentric longitude or right ascension, if JV denotes, in the fonner 
case, the longitude, in the latter, the right ascension, of the pole of the plane of 
the distances x. Wherefore, we shall have 

X — X= J cos b cos a 
y — F= A cos ^ sin a 
z — Z ■=^ A ^va.h . 

The two first equations will give a and A co^h; the latter quantity (which must 
be positive) combined with the thhd equation, will give h and A. 

61. 

We have given, in the preceding articles, the easiest method of determining 
the geocentric place of a heavenly body with respect to the ecliptic or equator, 
either free from parallax or affected by it, and in the same manner, either free 
from, or affected by, nutation. In what pertains to the nutation, all the difference 
will depend upon this, whether we adopt the mean or true position of the equator; 
as in the former case, we should count the longitudes from the mean equinox, 
in the latter, from the true, just as, in the one, the mean obliquity of the ecliptic 
is to be used, in the other, the true obliquity. It appears at once, that the greater 
the number of abbreviations introduced into the computation of the coordinates, 
the more the preliminary operations which are required ; on which account, the 
superiority of the method above explained, of deriving the coordinates immedi- 
ately from the eccentric anomaly, will show itself especially when it is necessary 
to determine many geocentric jDlaces. But when one place only is to be com- 
puted, or very few, it would not be worth while to undertake the labor of calcu- 
lating so many auxiliary quantities. It will be preferable in such a case not to 
depart from the common method, according to which the true anomaly and radius 
vector are deduced from the eccentric anomaly; hence, the heliocentric place 



Sect. 2.] to position in space. 71 

with respect to tlie ecliptic ; hence, the geocentric longitude and latitude ; and 
hence, finally, the right ascension and declination. Lest any thing should seem 
to be wanting, we will in addition briefly explain the two last operations. 

62. 

Let X be the heliocentric longitude of the heavenly body, /? the latitude ; I the 
geocentric longitude, h the latitude, r the distance from the sun, J the distance 
from the earth ; lastly, let L be the heliocentric longitude of the earth, B the lat- 
itude, R its distance from the sun. When we cannot put ^ ^ 0, our formulas 
may also be applied to' the case in which the heliocentric and geocentric places 
are referred, not to the ecliptic, but to any other plane whatever ; it will only be 
necessary to suppress the terms longitude and latitude : moreover, account can 
be immediately taken of the parallax, if only, the heliocentric place of the earth 
is referred, not to the centre, but to a point on the surface. Let us put, moreover, 

rcos/3=r/, J cosb = J', EcosB=:B'. 

Now by referring the place of the heavenly body and of the earth in space to 
three planes, of which one is the ecliptic, and the second and third have their 
poles in longitude JV and JV-\- 90°, the following equations immediately present 
themselves : — 

/ cos {l — JV) — R' cos [L — N)= A' cos {I — N) 
r' sin {l — N) — R' sin {L — N)^ A' sin {l—N) 
r' tan {-i — R' tan B =z J' tan h, 

in which the angle N is wholly arbitrary. The first and second equations will 
determine directly I — N and z/', whence h will follow from the third ; from h 
and J' you will have A. That the labor of calculation may be as convenient as 
possible, we determine the arbitrary angle N in the three following ways: — 
I. By putting iV= X, we shall make 



and I ^ 



^sin(^ — Z) = P, -^cos(l — Z) — 1= ^, 



72 EELATIONS PERTAINING SIMPLY" [BoOK ]. 

p 

tan(^ — L) = -^ 
A' P Q 



R' sin (l—L)~ cos (^ — L) 



tanJ=: 



-, tan fl — tan B 



n. By putting N^ ?., we shall make 

^•sin(2. — X) = P, l_:^cos(X — Z)=^, 

and we shall have, 

P 

tan (^ — V) 



Q 
A' P Q 



r' &m{l — l) cos {I— I) 

tan^: 



7?' 

tan /3 r tan B 



m. By putting iV= ^ (^ + ^)j ^ and J' will be found by means of the 
equations 

tan {l—^{l-{. L)) = ^^±|! tan ^{l—L) 

., (/ + iy)sin^(X — Z) {r' — R')Q.o%\{}. — L) 

~ sin(Z— 1(;L + X)) "~ cos(;-i(;i + Z)) ' 
and afterwards h, by means of the equation given above. The logarithm of the 
fraction 

t'^R' 
r' — R' 

is conveniently computed if -j is put = tan C, whence we have 

:-^ = tan(45° + c:). 

In this manner the method IH. for the determination of I is somewhat shorter 
than I. and 11.; but, for the remaining operations, we consider the two latter 
preferable to the former. 



Sect. 2.] 



TO POSITION IN SPACE. 



73 



63. 

For an example, we continue further the calculation carried to the helio- 
centric place in article 51. Let the heliocentric longitude of the earth, 
24:° 19' 4:9" M = L, and log J? = 9.9980979, correspond to that place; we put 
the latitude =0. We have, therefore, l — L = —17°24'20".Q7,logB' = B, 
and thus, according to method IL, 

log^ .... 9.6729813 log(l— ^) .... 9.6526258 

- log sin (^ — Z) . 9.4758653^ l—Q— 0.4493925 

^ logcos(X — Z) . 9.9796445 Q= 0.5506075 

--logP . . . . 9.1488466 w 
log^ .... 9.7408421 

Hence i—X = — 14^21' 6".75 whence 1= 352°34'22".23 
log^ . . . . 9.7546117 whence log z/' . . . 0.0797283 
log tan/? . . . 8.8020996 w log cos ^ 9.9973144 

logtanJ . . . 9.0474879?? log J 0.0824139 

b = — 6°2r55".07 

According to method HI., from log tan ^ = 9.6729813, we have C = 25°13'6''.31, 
and thus, 

log tan (45° + . • . 0.4441091 

logUn ^l — L) . . . 9.1848938JZ 

logtan(^— U — ^Z) . 9.6290029W 

l—n — iL=:— 23° 3'16".79 I whence^=:352°34'22'^225. 
H+iX:= 15 37 39.015] 

64. 

We further add the following remarks concerning the problem of article 62. 
I. By putting, in the second equation there given, 

JSr=l, JSrz=zL, N=il, 
10 



74 RELATIONS PERTAINING SIMPLY [BoOK I. 

there results 

Ii'sm{X — Z) = J'sm{l—X) 
r'sm{X—L) = //'sm{l—L) 
rsm{l — 1}=E' sin {I — X) . 

The first or the second equation can be conveniently used for the proof of the 
calculation, if the method I or II. of article 62 has been employed. In our 
example it is as follows : — 

logsin(X — X) . . . 9.4758653 ;z /— Z = — 3r45'26".82 

log4 9.7546117 



9.7212536^2 
logsin(?— X) . . . 9.7212536 n 

n. The sun, and the two points in the plane of the ecliptic which are the 
projections of the place of the heavenly body and the place of the earth form a 
plane triangle, the sides of which are J', B', /, and the opposite angles, either 
l—L,l—l, 180° — ^+X, or L — l, I— I, and im°'—L-\-l; from this the 
relations given in I. readily follow. 

III. The sun, the true place of the heavenly body in space, and the true place 
of the earth will form another triangle, of which the sides will be J, R, r : if, 
therefore, the angles opposite to them respectively be denoted by 

S,T, IW — S—T, 
we shall have 

&mS sinT sin {S^ T) 

A E r '' 

The plane of this triangle will project a great circle on the celestial sphere, in 
which will be situated the heliocentric place of the earth, the hehocentric place 
of the heavenly body, and its geocentric place, and in such a manner that the 
distance of the second from the first, of the third from the second, of the third 
from the first, counted in the same direction, will be respectively, S,T, 8 -\- T. 

IV. The following differential equations are derived from known differential 
variations of the parts of a pla,ne triangle, or with equal facihty from the formu- 
las of article 62: — 



Sect. 2.] to position m space. 75 

J, / cos 5 sin 5 sin (X — Ji i r' cos'b ,^ , cos^b ., ^ /i 7\ i^ i\ -i i 

d^= ^— ^ ^d^ + -,^^d/3 + -^^(tafl/3 — cos(2. — Z)taii^)dr', 

in which the terms which contain dr dA' are to be multiplied by 206265, or the 
rest are to be divided by 206265, if the variations of the angles are expressed in 
seconds. 

Y. The inverse problem, that is, the determination of the hehocentric from 
the geocentric place, is entirely analogous to the problem solved above, on which 
account it would be superfluous to pursue it further. For all the formulas of 
article 62 answer also for that problem, if, only, all, the quantities which relate to 
the heliocentric place of the body being changed for analogous ones referring to 
the geocentric place, L -j- 180° and — B are substituted respectively for L and B, 
or, which is the same thing, if the geocentric place of the sun is taken instead of 
the heliocentric place of the earth. 

65. 

Although in that case where only a very few geocentric places are to be 
determined from given elements, it is hardly worth while to employ all the 
devices above given, by means of which we can pass directly from the eccentric 
anomaly to the geocentric longitude and latitude, and so also to the right ascen- 
sion and declination, because the saving of labor therefrom would be lost in 
the preliminary computation of the multitude of auxiliary quantities; still, the 
comljination of the reduction to the ecliptic with the computation of the geocen- 
tric longitude and latitude will afford an advantage not to be despised. For if the 
ecliptic itself is assumed for the plane of the coordinates s, and the poles of 
the planes of the coordinates x,y, are placed in S2, 90° -[- ^, the coordinates are 
very easily determined without any necessity for auxiliary quantities. We have, 



a; = r cos u 
y=. r cos^sin^« 
s = r sin i sin u 



X=E'cos{L—Q) 
Y=B'sm{L—Q) 
Z=i R' tan B 



x — X=J'cos{l—i 
z — Z=J'idJib. 



76 



RELATIONS PERTAINING SIMPLY 



When B = (), tlien B'=R,Z^ 
solved as follows : — 

L 



[Book I. 
According to these formulas our example is 



213°12'0'^32. 

logi^' . . . 
log cos (Z — 9,) 
log sin (X — Q,) 



logr 0.3259877 

log cos M 9.9824141 w 

log sin M 9.4454714« 

log a; 0.3084018 w logX 9.9207006 w 



9.9980979 

9.9226027?z 

9.7384353W 



log r sin ■ 
log cos / 
log sin i 



9.7714591^ 

9.9885266 

9.3557570 



'y 



9.7599857^2 
9.1272161W 



log^ 

Hence follows 

\ogX^—X) . . . 0.0795906 w 

\og{y — Y) . . . 8.4807165W 
whence (/— a) = 18r26'33".49 

\ogJ' ...... 0.0797283 

log tan 5 9.0474878 « 



logZ 9.7365332re 

Z= 






352°34'22".22 
— 6 21 65.06 



The right ascension and declination of any point whatever in the celestial 
sphere are derived from its longitude and latitude by the solution of the spherical 
triangle which is formed by that point and by the north poles of the ecliptic and 
equator. Let e be the obhquity of the ecliptic, / the longitude, h the latitude, a 
the right ascension, d the declination, and the sides of the triangle will be e, 
90° — b, 90° — d ; it wiU be proper to take for the angles opposite the second 
and tMrd sides, 90° -\-a, 90° — /, (if we conceive the idea of the spherical triangle 
in its utmost generahty) ; the thhd angle, opposite e, we will put= 90° — E. "We 
shall have, therefore, by the formulas, article 54, 



Sect. 2.] TO POSITION in space. 77 

sin (45° — H) sin i (^ -f a) = sin (45° + I /) sin (45° — I (e + h)} 
sin (45° — ^ d) cos i{i;-\-a) = cos (45° -f- h I) cos (45° — i{e — b)) 
cos (45°— hd)sm^:E—a) = cos (45° + i I) sin (45° — ^ (e — b)) 
cos (45°— i (J) cos ^i:—a) = sin (45° + i I) cos (45° — i (e + b)) 

The first two equations wUl give ^ {IJ-\- a) and sin (45° — 2 d); the last two, 
^{U — a) and cos (45° — id); from i(^-f-^) ^^^ 2 (^ — a) will be had a, and, 
at the same time, U; from sin (45° — ^ (5") or cos (45° — I d), the agreement of 
which will serve for proving the calculation, will be determined 45° — id, and 
hence d. The determination of the angles i [U -\- a), i [E — a) by means of 
their tangents is not subject to ambiguity, because both the sine and cosine of the 
angle 45° — id must be positive. 

The differentials of the quantities a, d, from the changes of I, b, are found 
according to known principles to be, 

a = ^—dl ^ab 

cos cos 

did ^ COS E cosb dl-\-siia.E db. 

67. 

Another method is required of solving the problem of the preceding article 
from the equations 

cos 6 sin ^ = sin e tan b -\- cos I tan a 

sin d = cos £ sin 5 -|- sin t cos ^ sin / 
cos ^ cos / = cos a cos (^ . 

The auxiliary angle 6 is determined by the equation 

, . tan S 

tan^= -r-^, 
and we shall have 

tan«=^^^^^+^^ 

COSP 

tan d = sin « tan (e -|- ^) , 

to which equations may be added, to test the calculation, 

R cos 5 COS Z r> COS (s -\~ 6) cos b sin I 

coso = -or COSO:= — / . . 

cos a cos sin a 



78 RELATIONS PERTAINING SIMPLY [BoOK 1. 

This ambiguity in the determination of a by the second equation is removed by 
this consideration, that cos a and cos I must have the same sign. 

This method is less expeditious, if, besides a and ^, E also is requned : the most 
convenient formula for determining this angle will then be 

-r, sin £ COS a sin £ cos I 

cosJE:= ^— = — -. 

cos COS 

But JE cannot be correctly computed by this formula when + cos U differs but 
little from imity ; moreover, the ambiguity remains whether JE should be taken 
between and 180°, or between 180° and 360°. The inconvenience is rarely 
of any importance, particularly, since extreme precision in the value of U is not 
required for computing differential ratios j but the ambiguity is easily removed 
by the help of the equation 

cos b cos d sin ^ = cos e — sin J sin d, 

which shows that U must be taken between and 180°, or between 180° and 
360°, according as cose is greater or less than sin i sin^ : this test is evidently not 
necessary when either one of the angles h, d, does not exceed the limit 66° 32' ; 
for in that case sin U is always positive. Finally, the same equation, in the case 
above pointed out, can be appHed to the more exact determination of U, if it 
appears worth while. 

68. 

The solution of the inverse problem, that is, the determination of the longi- 
tude and latitude from the right ascension and declination, is based upon the same 
spherical triangle ; the formulas, therefore, above given, will be adapted to this 
purpose by the mere interchange of b with d, and of /with — a. It will not be 
unacceptable to add these formulas also, on account of their frequent use : 

According to the method of article 66, we have, 

sin (45° — i h) sin h (^— 7) = cos (45° + h a) sin (45°— ^ (^ +'^)) 
sin (45° — h h) cos k{E—l)=. sin (45^ + I a) cos (45° — ^e — d)) 
cos (45° —hb) sin h{E-\-l) = sin (45° + ^ «) sin (45° — I (e — d)) 
co8(45°— i^)cos|(^+/)=cos(45°+ia)cos(45° — i(£+^)). 



Sect. 2.] to position m space. 79 

As in the other method of article 67, we will determine the auxiliary angle ^ 
by the equation 

. y tan 5 

tan ^ = -. — , 

sin a' 

and we shall have 

, , cos (t — s) tana 

tan^= — 5^^ ^ 

cos Q 

tan b = sin Ha,n (^ — 6). 

For proving the calculation, may be added, 

, cos 5 cos a cos (C — e) cos d sin a 

COS b =^ J— = — ^ y^^^i • 

cos I cos f sm I 

For the determination of E, in the same way as in the preceding article, the fol- 
lowing equations will answer : — 

7-, sin £ cos a sin e cos I 

COSjE'= j—= -r— 

cos cos 

COS b COS ^ sin ^ := cos e — sin J sin d . 
The differentials of /, h, will be given by the formulas 

^ , sin^cosfi 1 I cos^ T ft 
dl= -,— d 0: -^ r d o 

cos ' cos 

db = — COS -E" COS ^ d « 4- sin -S" d d . 



We will compute, for an example, the longitude and latitude from the right 

ascension 355° 43' 45".30=r a, the declination — 8° 47' 25" = d^, and the obliquity 

of the echptic 23° 27' 59".26 = e. We have, therefore, 45^ + i a = 222° 51' 52".65, 

45° — i(e+(^) = 37°39'42".87, 45° — I (e — d") = 28°52'17".87; hence also, 

.ogcos(45°+2ia) . . 9.8650820W log sin (45° + i a) . . 9.8326803?2 

.ogsin(45° — |(e + d)) 9.7860418 logsin (45°— i (e — (^)) 9.6838112 

.ogcos(45°— i(e + ^)) 9.8985222 logcos(45° — ^e — (^)) 9.9423572 

.ogsin(45° — i^)sin^(^— ;) . . 9.6 511238 w 
ogsin(45° — f^)cos|(^— /) . . 9.7750375 w ■ 
whence ^ (^— /) = 216''56'5".39 ; log sin (45° —^b) = 9.8723171 



80 



RELATIONS PERTAIOTNG SIMPLY 



[Book L 



logcos(45°— i5)sm^^ + Z) . . 9.5164915 w 
logcos(45°— ^J)cos^(^+/) . . 9.7636042 w 
whence h{E^l)=^ 209° 30'49".94 : log cos (45° —hh) = 9.8239669. 

Therefore, we have E = 426° 26' 55".33, l = — T 25' 15".45, or, what amounts 
to the same thing, ^ == 66°26'55".33, /= 352°34'44".55; the angle 45° — i ^, 
obtained from the logarithm of the sine, is 48°10'58".12, from the logarithm of 
-the cosine, 48°10'58".17, from the tangent, the logarithm of which is their differ- 
ence, 48° 10' 68".14 ; hence ^ === — 6° 21' 56".28. 

According to the other method, the calculation is as follows : — 



log tan d 



9.1893062W 



log tan C .... 0.3173270 



C. log cos ^ 
log cos {t,— 
log tan a 



0.3626190 

9.8789703 
8.8731869« 






64°ir6".83 
40 49 7 .57 



log tan? 9.1147762 w 

1= 352°34'44".50 
log sin/ 9.1111232 w 

logtan(C — e) . . 9.9363874 



log tan ^ 9.0475106 ?z 

h= — 6°21'56".26. 

For determining the angle E we have the double calculation 

log sine 6.6001144 

log cos/ 9.9963470 

Clog cos ^ .... 0.0051313 

log cos ^ 9.6015927 



log sin e . . . 


. 9.6001144 


log cos a . . . 


, 9.9987924 


C. log cos J . . 


. 0.0026859 



log cos E 
whence E = 



. 9.6015927 
66°26'55".35. 



70. 

Something is stiU to be added concerning the paralhx and aberration, that 
nothing requisite for the computation of geocentric places maj be wanting. 
We have already described, above, a method, according to which, the place 
affected by parallax, that is, corresponding to any point on the surface of the 



Sect. 2.] to position m space. 81 

earth, can be determined directly with the greatest facility ; but as in the com- 
mon method, given in article 62 and the following articles, the geocentric place is 
commonly referred to the centre of the earth, in which case it is said to be free 
from parallax, it will be necessary to add a particular method for determining the 
parallax, which is the difference between the two places. 

Let the geocentric longitude and latitude of the heavenly body with reference 
to the centre of the earth be I, fi ; the same with respect to any point whatever 
on the surface of the earth be I, h ; the distance of the body from the centre of 
the earth, r ; from the point on the surface, //; lastly, let the longitude L, and the 
latitude B, correspond to the zenith of this point in the celestial sphere, and let 
the radius of the earth be denoted by R. Now it is evident that all the equations 
of article 62 will be applicable to this place also, but they can be materially 
abridged, since in this place R expresses a quantity which nearly vanishes in 
comparison with r and J. The same equations evidently will hold good if 1,1, L 
denote right ascensions instead of longitudes, and /?, h, B, declinations instead of 
latitudes. In this case / — 1, h — §, will be the parallaxes in right ascension and 
declination, but in the other, parallaxes in longitude and latitude. If, accord- 
ingly, R is regarded as a quantity of the first order, I — l,h — (i, J — r, will be 
quantities of the same order; and the higher orders being neglected, from the 
formulas of article 62 will be readily derived : — 

J , , R cos B sin (X — L) 

r cos ^ 

n. h-§=^-^^^-^{i^n(ioo^{l-L)-i^nB) 
m. J — r = — i? COS B sin /5 ( cotan j3 cos (X — L) -\- tan Bj . 
The auxiliary angle ^ being so taken that 

tan^ *^"^ 



cos {X — Ly 

the equations 11, and III. assume the following form : — 

-rj 7 o R COS B COS {I — L) sin ((3 — d) R sm B bvo. {B — d) 

^ r cos (9 r&md 

j-pr J i2cos-Bcos(^ — L)cos(§ — 6) ^sinjBcos(^ — d) 

cos (? sin 5 ' 

11 



82 



RELATIONS PERTAINING SIMPLY 



[Book 1. 



Further, it is evident, that in I. and II., in order that / — X and b — (i may be 
had in seconds, for B, must be taken the mean parallax of the sun in seconds ; 
but in in., for B, must be taken the same parallax divided by 206265". Finally, 
when it is required to determine in the inverse problem, the place free from 
parallax from the place affected by it, it will be admissible to use J, I, b, instead 
of r, I, fi, in the values of the parallaxes, without loss of precision. 

Example. — Let the right ascension of the sun for the centre of the earth 
be 220°46'44".65 = X,the declination,— 15° 49'43".94=^, the distance, 0.9904311 
=^ r : and the sidereal time at any point on the surface of the earth expressed 
in degrees, 78° 20' 38" = i:, the elevation of the pole of the point, 45°2r57" = ^, 
the mean solar parallax, 8".6 = R. The place of the sun as seen from this point, 
and its distance from the same, are required. 



logR 0.93450 

logcos^ 9.84593 

C.logr 0.00418 

Clog cos ^i .... 0.01679 
logsin(^ — Z) . . . 9.78508 
loga— ^) . . . . 0.58648 
1—1= 4-3".86 

1= 220°46'48".51 

log tan 5 0.00706 

logcos(X — X) . . . 9.89909W 
log tan ^ 0.10797m 

&= 127° Sr 0'' 

(i — &= —143 46 44 



logR 0.93450 

log sin ^ 9.85299 

C.logr . . . . .' . 0.00418 

Clog sin ^ 0.10317 

logsin(^ — ^) . . . 9.77152 w 

\og{b — ii) .... 0.66636 ^^ 
b — i^= — 4".64 

b= — 15°49'48".58 

log(^_/i}) .... 0.66636 « 
log cot (/i — ^) . . . 0.13522 

logr 9.99582 

logl" 4.68557 

log(r — z/) . . . . 5.48297"^ 
r — A= —0.0000304 

J= 0.9904615 



71. 

Tlie aberration of the fixed stars, and also that part of the aberration of com- 
ets and planets due to the motion of the earth alone, arises from the fact, that 
the telescope is carried along with the earth, whUe the ray of light is passing 



Sect. 2.] to position m space. 83 

along its optical axis. The observed place of a heavenly body (which is called 
the apparent, or affected by aberration), is determined by the direction of the 
optical axis of the telescope set in such a way, that a ray of light proceeding 
from the body on its path may impinge upon both extremities of its axis : but this 
direction differs from the true direction of the ray of light in space. Let us con- 
sider two moments of time t, t', when the ray of light touches the anterior ex- 
tremity (the centre of the object-glass), and the posterior (the focus of the object- 
glass) ; let the position of these extremities in space be for the first moment a, I ; 
for the last moment a',b'. Then it is evident that the straight line ab' is the true 
direction of the ray in space, but that the straight line ab or a'b' (which may be 
regarded as parallel) corresponds to the apparent place : it is perceived without 
dif&culty that the apparent place does not depend upon the length of the tube. 
The difference in direction of the right lines b'a, ba, is the aberration such as exists 
for the fixed stars : we shall pass over the mode of calculating it, as well known. 
This difference is still not the entire aberration for the wandering stars : the 
planet, for example, whilst the ray which left it is reaching the earth, itself 
changes its place, on which account, the direction of this ray does not correspond 
to the true geocentric place at the time of observation. Let us suppose the ray 
of light which impinges upon the tube at the time t to have left the planet at the 
time T ; and let the position of the planet in space at the time T be denoted by 
P, and at the time thj p ; lastly, let A be the place of the anterior extremity of 
the axis of the tube at the time T. Then it is evident that, — 

1st. The right line AP shows the true place of the planet at the time T', 

2d. The right line ap the true place at the time t ; 

3d. The right line ba or b'a' the apparent place at the time t or f (the differ- 
ence of which may be regarded as an infinitely small quantity) ; 

4th. The right line b'a the same apparent place freed from the aberration of 
the fixed stars. 

Now the points P, a, b', lie in a straight line, and the parts Pa, ab', will be 
proportional to the intervals of time t — T,f — t, if light moves with an uni- 
form velocity. The interval of time f — T is always very small on account of 
the immense velocity of light j within it, it is allowable to consider the motion 



84 RELATIONS PERTAINING SMPLY [BoOK I. 

of the earth as rectilinear and its velocity as uniform : so also A, a, a will lie in a 
straight line, and the parts Aa, ad will likewise be proportional to the intervals 
t — T, t' — t. Hence it is readily inferred, that the right lines AP, h'a' are paral- 
lel, and therefore that the first and third places are identical. 

The time t — T, within which the light traverses the mean distance of the 
earth from the sun which we take for unity, will be the product of the distance 
Pa into 493^ In this calculation it will be proper to take, instead of the dis- 
tance Pa, either PA or pa, since the difference can be of no importance. 

From these principles follow three methods of determining the apparent place 
of a planet or comet for any time t, of which sometimes one and sometimes 
another may be preferred. 

I. The time in which the light is passing from the planet to the earth may be 
subtracted from the given time ; thus we shall have the reduced time T, for which 
the true place, computed in the usual way, wiQ be identical with the apparent 
place for t For computing the reduction of the time t — T,\i is requisite to 
know the distance from the earth ; generally, convenient helps will not be want- 
ing for this purpose, as, for example, an ephemeris hastily calculated, otherwise it 
will be sufficient to determine, by a preliminary calculation, the true distance for 
the time t in the usual manner, avoiding an unnecessary degree of precision. 

II. The true place and distance may be computed for the instant t, and, 
from this, the reduction of the time t — T, and hence, with the help of the daily 
motion (in longitude and latitude, or in right ascension and declination), the re- 
duction of the true place to the time T. 

III. The heliocentric place of the earth may be computed for the time /; and 
the heliocentric place of the planet for the time T : then, from the combination 
of these in the usual way, the geocentric place of the planet, which, increased 
by the aberration of the fixed stars (to be obtained by a well-known method, or 
to be taken from the tables), will furnish the apparent place sought. 

The second method, which is commonly used, is preferable to the others, 
because there is no need of a double calculation for determining the distance, 
but it labors under this inconvenience, that it cannot be used except several 
places near each other are calcalatcd, or are known from observation ; otherwise 
it would not be admissible to consider the diurnal motion as o-iven. 



Sect. 2.] to position in space. 85 

The disadvantage with which, the first and third methods are incumbered, is 
evidently removed when several places near each other are to be computed. 
For, as soon as the distances are known for some, the distances next following 
may be deduced very conveniently and with sufficient accuracy by means of 
familiar methods. K the distance is known, the first method will be generally 
preferable to the third, because it does not require the aberration of the fixed 
stars ; but if the double calculation is to be resorted to, the third is recommended 
by this, that the place of the earth, at least, is retained in the second calculation. 

What is wanted for the inverse problem, that is, when the true is to be derived 
from the apparent place, readily suggests itself According to method I., you will 
retain the place itself unchanged, but will convert the time t, to which the given 
place corresponds as the apparent place, into the reduced time T, to which the 
same will correspond as the true place. According to method II., you, will retain 
the time t, but you will add to the given place the motion in the time t — iT, as 
you would wish to reduce it to the time t-^{t — T). According to the method 
ni., you will regard the given place, free from the aberration of the fixed stars, 
as the true place for the time T, but the true place of the earth, answering to 
the time t, is to be retained as if it also belonged to T. The utihty of the third 
method will more clearly appear in the second book. 

Finally, that nothing may be wanting, we observe that the place of the sun is 
affected in the same manner by aberration, as the place of a planet : but since 
both the distance from the earth and the diurnal motion are nearly constant, the 
aberration itself has an almost constant value equal to the mean motion of 
the sun in 493', and so = 20".25; which quantity is to be subtracted from the 
true to obtain the mean longitude. The exact value of the aberration is in the 
compound ratio of the distance and the diurnal motion, or what amounts to the 
same thing, in the inverse ratio of the distance ; whence, the mean value must be 
diminished in apogee by 0".34, and increased by the same amount in perigee. 
Our solar tables already include the constant aberration — 20".25 ; on which 
account, it will be necessary to add 20".25 to the tabular longitude to obtain the 
true. 



86 RELATIONS PERTAINING SBIPLY [BoOK 1. 

72. 

Certain problems, which are in frequent use in the determination of the orbits 
of planets and comets, will bring this section to a close. And first, we will revert 
to the parallax, from which, in article 70, we showed how to free the observed 
place. Such a reduction to the centre of the earth, since it supposes the distance 
of the planet from the earth to be at least approximately known, cannot be made 
when the orbit of the planet is wholly unknown. But, even in this case, it is pos- 
sible to reach the object on account of which the reduction to the centre of the 
earth is made, since several formulas acquire greater simplicity and neatness 
from this centre lying, or being supposed to lie, in the plane of the ecliptic, 
than they would have if the observation should be referred to a point out of the 
plane of the ecliptic. In this regard, it is of no importance whether the obser- 
vation be reduced to the centre of the earth, or to any other point in the plane 
of the ecliptic. Now it is apparent, that if the point of intersection of the 
plane of the ecliptic with a straight line drawn from the planet through the true 
place of observation be chosen, the observation requires no reduction whatever, 
since the planet may be seen in the same way from all points of this line:* where- 
fore, it will be admissible to substitute this point as a fictitious place of observa- 
tion instead of the true place. We determine the situation of this point in the 
following manner : — 

Let X be the longitude of the heavenly body, /5 the latitude, J the distance, 
all referred to the true place of observation on the surface of the earth, to 
the zenith of which corresponds the longitude /, and the latitude b ; let, more- 
over,- 7t be the semidiameter of the earth, L the heliocentric longitude of the cen- 
tre of the earth, B its latitude, B its distance from the sun ; lastly, let X' be the 
hehocentric longitude of the fictitious place, li' its distance from the sun, J -\-d 



* If the nicest accuracy should be wanted, it would be necessary to add to or subtract from the given 
time, the interval of time in which light passes from the true place of observation to the fictitious, or from 
the latter to the former, if we are treating of places affected by aberMion : but this difference can 
scarcely be of any importance unless the latitude should be very small. 



Sect. 2.] to position in space. 87 

its distance from the heavenly Ibody. Then, TV denoting an arbitrary angle, the 
following equations are obtained withaut any difficulty : — 

R' cos {L'—N) -f d cos (i cos {I —N) =EcosB cos {L—N) + tt cos 3 cos {l—N) 
R' sin {L'—N) + d cos {i sin {I —N) = R cos B sin {L—N) -f tt cos ^ sin {l—N) 

d mi ^ =^ R sin B -\- n mil . 
Putting, therefore, 

L (i? sin ^ -|- ^ ^i'^ ^) cotan ^ =fi, 
we shall have 

n. R' cos {L'—N) = R cos ^ cos {L —N)-\- n cos h cos {l—N)—fi cos (^ — iV) 
m. ^' sin {L' —N) = RcosB sin (Z —N) + tt cos ^ sin {l—N) —^ sin {I —N) 
IV. (^ = -^,. 

cos p 
From equations 11. and IH., can be determined R' and L', from IV., the inter- 
val of time to be added to the time of observation, which in seconds will be 
= 493 d. 

These equations are exact and general, and will be applicable therefore when, 
the plane of the equator being substituted for the plane of the ecliptic, L, L', I, X, 
denote right ascensions, and B, h, (i declinations. But in the case which we are 
specially treating, that is, when the fictitious place must be situated in the eclip- 
tic, the smallness of the quantities B, it, LI — L, still allows some abbreviation of 
the preceding formulas. The mean solar parallax may be taken for n ; B, for 
sin B ; 1, for cos B, and also for cos {L' — L)-, L' — L, for sin {L' — L). In this 
way, making N=^ L, the preceding formulas assume the following form : — 
I. jU, = {RB -\- n sin h) cotan /3 
n. ^' = 7? -|- TT cos ^ cos (/ — L) — lit cos (X — L) 

j-rj jf J 7rcos5sin(Z — X) — jit sin (i — X) 

R' 

Here B, n, L' — L are, properly, to be expressed in parts of the radius ; but it is 

evident, that if those angles are expressed in seconds, the equations I., III. can be 

retained without alteration, but for 11. must be substituted 

TV jy j^ 7t COS b COS {I — L) — n cos Q. — L) 

IC—M-\ 206265" * 



RELATIONS PERTAINING SBIPLY 



[Book I. 



Lastly, in the formula HI., R may always be used in place of the denominator i?' 

without sensible error. The reduction of the time, the angles being expressed 

in seconds, becomes 

493'. fi 
206265". cos /3' 



log; 7T . . . . 


. . 0.93450 


log sin 5 . . . 


. . 9.86330 


log n smb . . 


. . 0.79780 



73. 

Umm2)le. — Let X r= 354° 44' 54", /5 = — 4° 59' 32", ^=24° 29', ^ =: 46° 63', 
L'= 12° 28' 54", B = ^ 0".49, B = 0.9988839, n = 8".60. The calculation is as 
follows : — 

logE 9.99951 

log^ 9.69020 

\ogBE 9.68971 

Hencelog(^i?-l-7rsinJ) . 0.83040 
logcotan^ .... 1.05873 w 

logjiA 1.88913?^ 

logTT 0.93450 

log cos 5 9.83473 

logl" 4.68557 

log cos (/—X) . . . 9.99040 

5.44520 

number + 0.0000279 

Hence is obtained B' = B-{- 0.0003856 = 0.9992695. Moreover, we have 



log|i* 1.88913 jz 

logr 4.68557 

logcos(X — Z) . . . 9.97886 
6.55356?i 
number — 0.0003577 



logTTcos^ 0.76923 

logsin(/ — X) . . . 9.31794 
C.logi^' 0.00032 

0.08749 
number 4- r.22 



log lit 1.88913 w 

logsin(X — Z) . . 9.48371W 
C.logi?' 0.00032 



1.37316 



number + 23".61 



Sect. 2.] to position in space. 89 

Whence is obtained L'=^L — 22".39. Finally we have 

log;i* 1.88913W 

Clog 206265 .... 4.68557 

log 493 2.69285 

Clog cos/? 0.00165 

9.26 920 w, 
whence the reduction of time = — 0M86, and thus is of no importance. 

74. 

The other problem, to deduce the heliocentric place of a heavenly lody in its orlit 
from the geoceiitric place and the sanation of the plane of the orhit, is thus far similar to 
the precediugj that it also depends upon the intersection of a right line drawn 
between the earth and the heavenly body with the plane given in position. The 
solution is most conveniently obtained from the formulas of article 65, where the 
meaning of the sjnnbols was as follows : — 

L the longitude of the earth, R the distance from the sun, the latitude B we 
put =0, — since the case in which it is not =: 0, can easily be reduced to this by 
article 72, — whence R' = R, I the geocentric longitude of the heavenly body, b 
the latitude, /I the distance from the earth, r the distance from the sun, u the 
argument of the latitude, Q, the longitude of the ascending node, i the inclination 
of the orbit. Thus we have the equations 

I. rcosM — ^cos(X — 9>)=zJ coBh c,ob{1 — 9,) 
n. r cos z sin ?{ — i? sin (Z — Q ) = // cos b sin [I — 9, ) 
ni. r sin i sin ^f = z/ sin J . 

Multiplying equation I. by sin (Z — Q>) sin b, 11. by — cos (Z — -9,) sin b, IH. by 
— sin (Z — I) cos b, and adding together the products, we have 

cos u sin [L — Q, ) sin J — sin u cos i cos (X — ^ ) sin J — sin if sin «' sin [L — I) cos J = 0, 

whence 

lY. i^nu= sin(L-Q)smh 



cos i cos (X — ^ ) sin 5 -[- sin i sin (L — ^) cos i 

12 



90 RELATIONS PERTAINING SIMPLY [BoOK I. 

Multiplj-ing likewise I. by sin [l — 9,), 11. by — cos [l — Q), and adding together 
the products, we have 

^j- -ffi sin (L — I) 

sin u cos i cos (I — Q,) — cos m sin (/ — Q)' 

The ambiguity in the determination of u by means of equation IV., is removed 
by equation HI., which shows that it, is to be taken between and 180°, or be- 
tween 180° and 360° according as the latitude b may be positive or negative ; 
but if J = 0, equation V. teaches us that we must put ic = 180°, or u = 0, accord- 
ing as sin [L — I) and sin {I — Q) have the same or different signs. 

The numerical computation of the formulas IV. and V. may be abbreviated in 
various ways by the introduction of auxiliary angles. For example, putting 

tan5c^(Z-^)^^^^^ 
sin (Z — I) ' 

we have 

sin {A~\-i) ' 

putting 

tan i sin (i — I) , -r^ 

.j.^ - / = tan B, 

cos{L—Q) ' 

we have 

sin (iJ-j-o) cos I 

In the same manner the equation V. obtains a neater form by the introduction 
of the angle, the tangent of which is equal to 

COS z tan u, or — ^ — : — -. 
' cos^ 

Just as we have obtained formula V. by the combination of I., 11., so by a combina- 
tion of the equations IT., LI., we arrive at the following : — 

^^ igsin(Z-a) . 

sin u (cos i — sin i sin (l — £1) cotan b) ' 

and in the same manner, by the combination of equations I., IH., at this ; 

i?cos(Z— g2) 



sm u sill t cos 



(^—S^) cotan J' 



Sect. 2.] to position in space. 91 

both of whicli, in the same manner as Y., may be rendered more simple by the 
introduction of auxiliary angles. The solutions resulting from the preceding 
equations are met with in Von Zach Monatliche Correspondens, Vol. V. p. 540, col- 
lected and illustrated by an example, wherefore we dispense with their further 
development in this place. If, besides u and r, the distance J is also wanted, it 
can be determined by means of equation III. 

75. 

Another solution of the preceding problem rests upon the truth asserted in arti- 
cle 64, in., — that the heHocentric place of the earth, the geocentric place of the 
heavenly body and its heliocentric place are situated in one and the same great 
circle of the sphere. In fig. 3 let these places be respectively T, G, H; further, 
let 9> be the place of the ascending node ; 9>T, 9,11, parts of the ecliptic and 
orbit ; GP the perpendicular let fall upon the ecliptic from G, which, therefore, 
wiUbe=^. Hence, and from the arc PT=i/ — ^will be determined the angle T 
and the arc TG. Then in the spherical triangle 9, HT are given the angle 9> = i, 
the angle T, and the side 9>T=^L — 9>, whence will be got the two remaining 
9, R= u and TK Finally we have HG = TG— TH, and 

R sin TG . R sin TH 



&mHG ' smHG ' 



76. 



In article 52 we have shown how to express the differentials of the heliocen- 
tric longitude and latitude, and of the curtate distance for changes in the argu- 
ment of the latitude u, the inchnation i, and the radius vector r, and subsequently 
(article 64, IV.) Ave have deduced from these the variations of the geocentric 
longitude and latitude, I and b : therefore, by a combination of these formulas, d I 
and d^ will be had expressed by means of d^f, d^', dS2, dr. But it wiU be worth 
while to show, how, in this calculation, the reduction of the heliocentric place 
to the ecliptic, may be omitted in the same way as in article 65 we have 
deduced the geocentric place immediately from the heliocentric place in orbit. 
That the formulas may become more simple, we will neglect the latitude of 



92 RELATIONS PERTAINING SIMPLY [BoOK I. 

the earth, which of course can have no sensible effect in differ'ential formulas. 
The following formulas accordingly are at hand, in which, for the sake of brevity, 
we write w instead of/ — Q,, and also, as above, J' in the place of z^ cos J. 
J' cos (0 = r cos u — M cos (X — S^ ) z= ^ 
J' sinb) =:^rcosismu — i?sin(Z — Q,)-=fj 
A' tan 3 = r sin i sin u^=lL,\ 
from the diflferentiation of which result 

cos (o.dJ' — z/'sin w.dto = d^ 
sin to . d ^' -j- ^' cos to . d w = d?^ 

tan5.dz/'4-^,d5==dC. 

' cos 
Hence by elimination, 

— sin w . d J -1- cos oi.dti 

0)^ '-^ 

A 

■, 7 — COS to. sin 5. d 5 — sin wsin J.dj/-)- cosS.d^ 

A 

K in these formulas, instead of ^, r], t, their values are substituted, dto 

and db will appear represented by dr, du, di, dQ, ; after this, on account of 

d/=dto-|-dS, the partial differentials of I and b will be as follows : — 

<dZ> 



sm to cos II, -j- cos to sm u cos i 



il. — I ^^ I = sm to sm ?f + cos o) cos u cos ; 

TTT ^'/<iZ\ . . . 

ill. — ( — ■ ) = — COS 0) sm u sm i 

^- (^)=l + |cos(X-g2-to) = l+|cos(Z-0 

Y. J ( — j = — cos to cos u sin b — sin to sin ?< cos i sin b -\- sin u sin ^ cos b 

VI. —(—) ^ cos to sin u sin h — sin to cos u cos ^' sin J 4- cos m sin i cos b 

r \au/ ' 

Vn. — (t-.) = sin (i) sin ic sin i sin b -j- sin u cos i cos b 

Vm. ;»-(^ q) = sin 5 sin (Z — Q — to) := sin J sin (L — I). 



Sect. 2.] to position in space. 93 

The formulas IV. and "VILL. already appear in tlie most convenient form for cal- 
culation ; but the formulas L, HL, Y., are reduced to a more elegant form by 
obvious substitutions, as 

m.* (^) =— cos w tan h 

v.* (j-) = — -^^^^{^ — ^sinJ = — — ,cos(Z — ^)sinJcosJ. 

Finally, the remaining formulas 11., VI., VII., are changed into a more simple form 
by the introduction of certain auxiliary angles : which may be most conveniently 
done in the following manner. The auxiliary angles M, N, may be determined 
by means of the formulas 

tan Jf = — % , tan iV= sin to tan i = tan Jf cos to sia ^. 

COSi ' 

Then at the same time we have 

cos^M 1 4- tan^ N cos^ i 4- sin^ to sin^ i « 

cos2J\r — l+tan^if ~ 'cos^i + tan^w — ^"^ »" • 

now, since the doubt remaining in the determination oi M, N,hj their tangents, 
may be settled at pleasure, it is evident that this can be done so that we may 

have 

cos M 1 

^ = H- cos to, 

cos iv ' ' 

and thence 

sa\N ... 

-T-Tj= = + smz. 

sin if ' 

These steps being taken, the formulas IE., YI., VII., are transformed into the fol- 
lowing : — 

TT* (^^\ »* sin ft) COS (iHf — m) 

^' Wu) ~ ^' sin if 

VI.* (i^) = ^(j^o^ w sin 2 cos (IT— m) cos (iV — h)-\-mi{M — u) sin {N — h)) 

VTT * (^ ^\ ^ ®^^ " ^''^ * *^°^ ("^ — ^) 

\d 1/ ^ cos i\r 



94 RELATIONS PERTAINING SIMPLY [BoOK I. 

These transfonnations, so far as the formulas 11. and VH. are concerned, will detain 
no one, but in respect to formula VI., some explanation will not be superfluous. 
From the substitution, in the first place, of M — ( Jf — u) for u, in formula YI., 
there results 

— (^j = cos [M — u) (cos (a sin Jlf sin h — sin w cos/cos Jf sin h -\- sin^ cos J/cos h) 
— sin [M — u) (cos to cos Jf sin5 -[- sin m cos i sin J/ sin h — sin i siniJ/cos h). 
Now we have 

cos w sin M=z cos^ i cos w sin M-\- sin^ i cos w sin M 
= sin CO cos i cos M-\- sin^ i cos w sin M; 

whence the former part of that expression is transformed into 

sin i cos {M — u) (sin i cos la sin iHf sin b -\- cos Jlf cos h) 
= sin i cos {M — u) (cos w sin iV^sin l)-{-coBOi cos iVcos 5) 

= cos oi sin 2 cos [M — u) cos (iV — h) . 
Likewise, 

cos JV^ cos^ CO cos iV-j- sin^ co cos 1^=^ cos co cos 31 -\- sin co cos i sin iHf ; 
whence the latter part of the expression is transformed into 

— sin {M — u) (cos iV^sin b — sin iV^cos b) = sin {31 — it) sin (iV — b) . 

The expression YI.* follows directly from this. 

The auxiliary angle 3f can also be used in the transformation of formula I., 
which, by the introduction of 31, assumes the form 

\dr/ A' sin M 

from the comparison of which with formula I * is derived 

— E sin [L — I) sin 3f= r sin co sin ( Jf — u) ; 
hence also a somewhat more simple form may be given to formula 11.*, that is, 

That formula VI.* may be still further abridged, it is necessary'- to introduce 
a new auxiliary angle, which can be done in two ways, that is, either by putting 



Sect. 2.] TO position in space. 95 

, 7-, tan (M—u) , ^ tan (N'—h) 

tan P = — ^ — ^-^, or tan Q = — ^^ — ^^ ; 

cos CO sin I ' ^ cos to sin i ' 

from which results 

VT * * {—\ ^ ^^^ {M— u) cos {N— h — P) r sin {N— b) cos {M— u — Q) 

Vdw/ ^sinP ^sin^ 

The auxiliary angles M,N,P, Q, are, moreover, not merely fictitious, and it would 
be easy to designate what may correspond to each one of them in the celestial 
sphere ; several of the preceding equations might even be exhibited in a more 
elegant form by means of arcs and angles on the sphere, on which we are less 
inclined to dwell in this place, because they are not sufficient to render superflu- 
ous, in numerical calculation, the formulas above given. 

77. 

What has been developed in the preceding article, together with what we 
have given in articles 15, 16, 20, 27, 28, for the several kinds of conic sections, 
wiU furnish aU which is required for the computation of the dififerential varia- 
tions in the geocentric place caused by variations in the individual elements. 
For the better illustration of these precepts, we will resume the example treated 
above in articles 13, 14, 51, 63, 65. And first we will express dl and d^ in terms 
of 6.r, du, dz, do,, according to the method of the preceding article; which cal- 
culation is as foUows : — 

log tan 0) . 8.40113 log sin w . 8.40099 ?z logtsin (M—u) 9.41932 re 
log cos 2 . 9.98853 log tan ^ . 9.36723 log cos to sin 2 . 9.35562 w 

log tan If. 8.41260 logtaniV^ . 7.76822re log tan P . . 0.06370 
M = r28'52'' iV^=179°39'50" P= 49° 11' 13" 

M—u=lQ^17 8 JV—b=lSQ 145 JSr—l — P= 136 50 32 



96 



RELATIONS PERTAINING SIMPLY 



[Book I. 



I* 

log sm(Z-^) 9.72125 
logi? . . 9.99810 
Clog J' . 9.92027 



(*) . . 
C. log r 



9.63962 
9.67401 



log(|-^) . 9.31363 

IV. 

log^ . . 9.91837 
log cos{Z—I) 9.92956 



n.** 

(*)... 9.63962 
log cot(Jf—M) 0.58068^2 



m.* 



M^ 



0.22030 



. 9.84793 

vn.* 

log rsin^tcos^ 9.75999m 
log cos(iV— J) 9.99759 « 
C.log^. . 9.91759 
C.loa-cosiV 0.00001 w 



Y* 

(**)... 9.84793 

log sin 3 cos J 9.04212m 

C.logr . . 9.67401 



H(^!) 



.56406 



vm. 

(*)... 9.63962 
log sin 5 cos J 9.04212 m 



M!^) 



8.68174m 



log cos (0 
log tan h 



9.99986m 
9.04749m 



lo=K^) 



9.04735w 



VI.** 

log^ ... 0.24357 
log sin (iif—w) 9.40484 
log cos(i\^— ^P) 9.86301m 
Clog sin P . 0.12099 



M^ 



9.63241m 



0.11152 d^-f 1.70458 d 52 
0.47335 de — 0.04805 dS2. 



log(l-^) . 9.67518m 

These values collected give 

d/=: + 0.20589 dr + 1.66073 du 
d^ = -f 0.03665 dr — 0.42895 d« • 

It will hardly be necessary to repeat here what we have often observed, namely, 
tbat either the variations d/, d^, dii, di, dQ, are to be expressed in parts of the 
radius, or the coefficients of dr are to be multiphed by 206265", if the former are 
supposed to be expressed in seconds. 

Denoting now the longitude of the perihelion (which in our example is 



Sect. 2.] to position in space. 97 

52°18'9''.30) by i7, and the true anomaly by v, the longitude in orbit will be 
?f -j- ^ = t' -|- 77", and therefore du =dv-{-dIT — dQ, which value being sub- 
stituted in the preceding formulas, dl and db will be expressed in terms of dr, 
dv, dIT, dQ, d^. Nothing, therefore, now remains, except to express dr and dv, ac- 
cording to the method of articles 15, 16, by means of the differential variations 
of the elliptic elements.* 

We had in our example, article 14, 

log^ = 9.90355^ log© 

log a 0.42244 



log— ..... 0.19290 
log cosy .... 9.98652 



log tan 9 .... 9.40320 
log sin y .... 9.84931 w 



log(^) .... 0.17942 T^TT 

^^^^^ log(f4). . . . 9.67495W 



log a 0.42244 

log cos 9 .... 9.98652 
log cosy .... 9.84966 



0.25862^2 



2 — ecos^= 1.80085 

ee= 0.06018 

1.74067 

log 0.24072 

log:-^ 0.19290 log^-^) 

log sin ^ . . . . 9.76634 ?2 

log(^) .... 0.19996 w 

Hence is collected 

dy = + 1.51154 dif — 1.58475 dcp 

dr = — 0.47310 d M— 1.81393 dg) -f- 0.80085 da ; 

which values being substituted in the preceding formulas, give 

d/= -f 2.41287 dM— 3.00531 dy + 0.16488 da -\- 1.66073 diZ 

— 0.11152 d^ + 0.04385 dg? 

dJ = — 0.66572 dilf + 0.61331 dcp-}- 0.02925 da — 0.42895 diT 

— 0.47335 d»-f 0.38090 dQ. 



* It will be perceived, at once, that the symbol M, in the following calculation, no longer expresses 
our auxiliary angle, but (as in section 1) the mean anomaly. 

18 



98 RELATIONS PERTAINING SIMPLY [BoOK I 

If the time, to which the computed place corresponds, is supposed to be 
distant n days from the epoch, and the mean longitude for the epoch is 
denoted by iV^ the daily motion by t, we shall have Jf = N-\-ni: — 77, and thus 
d Jf = diV-f" ^^^"^ — ^^- ^'^ our example, the time answering to the computed 
place is October 17.41507 days, of the year 1804, at the meridian of Paris : if, 
accordingly, the beginning of the year 1805 is taken for the epoch, then 
n=^ — 74.58493; the mean longitude for that epoch was 41°52'21".61, and the 
diurnal motion, 824''.7988. Substituting now in the place of dJtf its value in 
the formulas just found, the differential changes of the geocentric place, expressed 
by means of the changes of the elements alone, are as follows: — 

&l=z 2.41287 ^N— 179.96 dr — 0.75214 d77— 3.00531 dg) -f 0.16488 da 

— 0.11152 d^ + 0.04385 da, 
d^ = _ 0.66572 di\r-f 49.65 dr + 0.23677 d77-f 0.61331 d^ + 0.02935 da 
_ 0.47335 d^ + 0.38090 d 9, . 

K the mass of the heavenly body is either neglected, or is regarded as 
known, r and a will be dependent upon each other, and so either dr or da may 
be eliminated from our formulas. Thus, since by article 6 we have 

we have also 

dr g da 

in which formula, if d-r is to be expressed in parts of the radius, it will be neces- 
sary to express r in the same manner. Thus in our example we have 

logT- 2.91635 

logr 4.68557 

log I 0.17609 

Clog a .... 9.57756 

log^ 7.35557W, 

or, dT = — 0.0022676 da, and da = — 440.99 d-r, which value being substituted 
in our formulas, the final form at length becomes : — 



Sect. 2.] to position in space. 99 

. d^= 2.41287 dJV— 252.67 dT — 0.75214 d77— 3.00531 dcp 
— 0.11152 d/+ 0.04385 dg^, 
d3 = — 0.66572 di\^+ 36.71 dT + 0.23677 di7+ 0.61331 dtp 
— 0.47335 d^■+ 0.38090 dS. 
In the development of these formulas we have supposed all the differentials dl, 
db, dJV, dr, dIT, dcp, di, do, to be expressed in parts of the radius, but, mani- 
festly, by reason of the homogeneity of all the parts, the same formulas will 
answer, if all those differentials are expressed in seconds. 



THIRD SECTION. 

RELATIONS BETWEEN SEVERAL PLACES IN ORBIT. 



78. 

The discussion of the relations of two or more places of a heavenly body in 
its orbit as well as in space, furnishes an abundance of elegant propositions, such 
as might easily fill an entire volume. But our plan does not extend so far as to 
exhaust this fruitful subject, but chiefly so far as to supply abundant facilities for 
the solution of the great problem of the determination of unknown orbits from 
observations : wherefore, neglecting whatever might be too remote from our pur- 
pose, we will the more carefully develop every thing that can in any manner 
conduce to it. We will preface these inquiries with some trigonometrical propo- 
sitions, to which, since they are more commonly used, it is necessary more fre- 
quently to recur. 

I. Denoting by A, B, 0, any angles whatever, we have 

sm ^ sin ( C— ^) -[- sin ^ sin (^ — C') 4- sin C'sin (^ — ^) = 
cos^sin(C— ^) + cos^sm(iL — 6') + cosCsin(^ — ^) = 0. 

n. If two quantities p, P, are to be determined by equations such as 
p sin [A — P)=za 
psm{B — P) = b, 
it may generally be done by means of the formulas 

p sin {B — A) sin {IT— P) = h sin {H— A)~a sin {H— B) 
p sin [B — A) cos {R— P) = b cos (//— A) — a cos [H— B), 

in which H is an arbitrary angle. Hence are derived (article 14, 11.) the angle 
H — P, and p sin {B — A); and hence P and p. The condition added is gen- 
(100) 



Sect. 3.] relations betwee?^ several places in orbit. 101 

erallj that p must be a positive quantity, whence the ambiguity in the deter- 
mination of the angle II — -Pby means of its tangent is decided; but without 
that condition, the ambiguity may be decided at pleasure. In order that the 
calculation may be as convenient as possible, it will be expedient to put the arbi- 
trary angle H either •= Aox =^B or :=: ^ ( J. -j- B\ In the first case the equa- 
tions for determining P andj» will be 

jo sin {A — P) =a, 

In the second case the equations wUl be altogether analogous ; but in the third 
case, 

And thus if the auxiliary angle ^ is introduced, the tangent of which z=-, P wiU 
be found by the formula 

tan {hA^hB — P) = tan (45° + C) tan h {B—A), 
and afterwards p by some one of the preceding formulas, in which 

I (^ _ a) = COS (45° + 1) v/^, = ^^^^i^;±^ - ^^^^^^^ 

^ ^ ^ I -''V sin 2; sm4y/2 cos C V^ 

in. If p and P are to be determined from the equations 
jt?cos(^ — P) ■=^a, 
p(ios{B — P)=h, 
every thing said in 11. could be immediately applied provided, only, 90° -|- A 
W -\-B were written there throughout instead of A and B : that their use may 
be more convenient, we can, without trouble, add the developed formulas. The 
general formulas will be 

p sin {B — A) sin [H— P) = — h cos {H— A)-\-a cos [H— B) 
p sin {B — A) cos {H— P) = h sin {H— ^) — « sin {H— B) . 
Thus for 11=^ A, they change into 



102 RELATIONS BETWEEN SEVERAL [BoOK I. 

• I A Ti\ a COS (5 — A) — b 

pBm{A-P)= ,.,\^_^) 
pcos{A — P) = a. 
For ff= B, they acquire a similar form ; but for ^= \{A-\-B) they become 
;,sin(M+Ji?-P) = ,-^4^ 

^cos(M+ii?-P) = ,4y_^) , 

SO that the auxiliary angle t being introduced, of which the tangent =p it 
becomes 

tan (M + J ^ — P) = tan (C — 45°) cotan \{B — A). 
Finally, if we desire to determine jt? immediately from a and h without previ- 
ous computation of the angle P, we have the formula 

p m^[B — A) = sj {aa'-\-ll — 1 al (^Q%{B — A)), 
as well in the present problem as in 11. 

79. 

For the complete determination of the conic section in its plane, three things 
are required, the place of the perihehon, the eccentricity, and the semi-parameter. 
If these are to be deduced from given quantities depending upon them, there 
must be data enough to be able to form three equations independent of each 
other. Any radius vector whatever given in magnitude and position furnishes 
one equation : wherefore, three radii vectores given in magnitude and position are 
requisite for the determination of an orbit ; but if two only are had, either one 
of the elements themselves must be given, or at all events some other quantity, 
with which to form the third equation. Thence arises a variety of problems 
which we will now investigate in succession. 

Let r, /, be two radii vectores which make, with a right line drawn at pleasure 
from the sun in the plane of the orbit, the angles N, N', in the direction of the 
motion ; further, let 77 be the angle which the radius vector at perihehon makes 
with the same straight line, so that the true anomaHes N — IT, N' — 77 may 
answer to the radii vectores r, / ; lastly, let e be the eccentricity, and p the semi- 
parameter. Then we have the equations 



Sect. 3.] places in oebit. 103 

^==l_^ecos(i\^— 77) 

^=l-f gcos(i\^' — 77), 

from wMch, if one of the quantities p, e, 77, is also given, it will be possible to 
determine the two remaining ones. 

Let us first suppose the semi-parameter p to be given, and it is evident that 
the determination of the quantities e and 77 from the equations 

ecos(i\^— 77) = ^— 1 
gcos(i\r'_77) = ^— 1, 

can be performed by the rule of lemma m. in the preceding article. We have 
accordingly 

tan {N- 77) ^ cotan {N'-N) - , (^4^-g,_^ • 

tan {kNJr^N'-n) = ^^- '^ '"^'^'^ ^ fj^ . 
80. 

If the angle 77 is given, p and e will be determined by means of the equations 

rr' (cos {N— TI) — cos {N' — IT)) 



p 



r cos (iV— n) — r' cos (N' — 77) 



r cos {N — 77) — r' cos {K' — .77) * 
It is possible to reduce the common denominator in these formulas to the form 
a cos {A — 77), so that a and A may be independent of IT. Thus letting H de- 
note an arbitrary angle, we have 

r cos(iV— 77)— /cos (i\^'— 77) = (rcos (i\^— .S")— /cos(i\^'— 5^)) cos(^— 77) 

— (r sin [N—H) —/sin {N'—H)) sin {R—U) 
and so 

= a cos (A — 77), 

if a and A are determined by the equations 

rcos{JV—R) — rcos{N' — B:)=acos{A — ff) 



104 RELATIONS BETWEEN SEVERAL [BoOK I. 

In this way we have 

_ 2r'/sini(N'~IP)s\n(^N'Jri^'—n) 
-^ a cos (J. — IT) 



a cos {A — J7) ■ 
These formulas are especially convenient when p and e are to be computed for 
several values of 11 ; r, r, N, N' continuing the same. Since for the calculation 
of the auxiliary quantities a, A, the angle H may be taken at pleasure, it will be 
of advantage to put H=^ h (-^-|- J^'), by which means the formulas are changed 
into these, — 

(/ _ r) cos ^{N' — N) = — a cos (.1 — h N— h N') 
{r' -f r) sin ^{N' — N) = — a sin {A — h N— h iSf'), 
And so the angle A being determined by the equation 

tan {A — h N— i N') = ^^ tan ^ {W' — iV) , 

we have immediately 

_ cos (A — I IT— jN') 

^ cos i (xV — N) cos (A—n) ' 

im c 
method already frequently explained 



The computation of the logarithm of the quantity -p^- may be abridged by a 



81. 

If the eccentricity e is given, the angle 77 will be found by means of the 
equation 

C0S(7l 11)— ecos^iN'~W) ' 

afterwards the auxiliary angle A is determined by the equation 

tan (A — i N— i N') = ^I^ tan i {N' — N)! 

The ambiguity remaining in the determination of the angle A — 77 by its cosine 
is founded in the nature of the case, so that the problem can be satisfied by two 
different solutions ; which of these is to be adopted, and which rejected, must be 
decided in some other way ; and for this purpose the approximate value at least 



Sect. 3.] places in orbit. 105 

of n must be already known. After IT is found, ^; will be computed by the 
formulas 

JO == /■ (1 -f e cos (iV^— 77)) = / (1 + e cos (iV^' — 77)), 

or by this, 

2 >•/ e sin ^ {N ' — N) sin (^ N'^^ N— 11) 

82. 

Finally, let us suppose that there are given three radii vectores r, r, r", which 
make, with the right line drawn from the sun in the plane of the orbit at pleasure, 
the angles iV, iV', iV"". We shall have, accordingly, the remaining symbols being 
retained, the equations 

(L) l = lJ^eco8 {H— IT) 

f=:l+.cos(iV^"-77), 

from which p, IT, e, can be derived in several different ways. If we wish to 
compute the quantity jo before the rest, the three equations (I.) maybe multiphed 
respectively by sin {N" — N'\ — sin(iV" — N\ sin [N' — N\ and the products 
being added, we have by lemma I., article 78, 

sin {N" — N') — sin {N" — N)^ sin {W — N) 



p: 



\mi{N"—N') — ymi{N"—N)^~mi{N' — N) 



This expression deserves to be considered more closely. The numerator evidently 
becomes 

2 sin h {N" — N') cos h {N" — N') — 2mi h {N" — N') cos {^ N" -\- h N' — N) 
= 4 sin ^ {N" — N') sm i {N" — N) sin h {N' — N). 
Putting, moreover, 

r'r"%m{N" — N')=n, rr" Bm{N" — N)=^n', rr'Bm{N' — N) = n", 

it is evident that h n, h n' i n", are areas of triangles between the second and thu-d 
radius vector, between the first and third, and between the first and second. 

14 



106 RELATIONS BETWEEN SEVERAL [BoOK I. 

Hence it will readily be perceived, that in the new formula, 

__ 4sin 1 {N'" — N') sin ^ (¥" — J^) sin ^ (J^' — IP) . r / /' 
" n — n' -\- n" 

the denominator is double the area of the triangle contained between the ex- 
tremities of the three radii vectores, that is, between the three places of the 
heavenly body in space. When these places are httle distant from each other, 
this area will always be a very small quantity, and, indeed, of the third order, 
if N' — N, N" — N' are regarded as small quantities of the first order. Hence 
it is readily inferred, that if one or more of the quantities r, r, r", N, N', N", are 
affected by errors never so slight, a very great error may thence arise in the de- 
termination of j» ; on which account, this manner of obtaining the dimensions of 
the orbit can never admit of great accuracy, except the three heliocentric places 
are distant from each other by considerable intervals. 

As soon as the semi-parameter j» is found, e and 77 will be determined by the 
combination of any two whatever of the equations I. by the method of article 79. 

83. 

If we prefer to commence the solution of this problem by the computation 
of the angle 77, we make use of the following method. From the second of 
equations I. we subtract the third, from the first the third, from the first the sec- 
ond, in which manner we obtain the three following new equations : — 
1 1 

sm(iN-\-iN" — n) 



Any two of these equations, according to lemma H., article 78, will give 77 and -, 
whence by either of the equations (I.) will be obtained likewise e and p. If we 
select the third solution given in article 78, II., the combination of the first equa- 





1 


1 




2 


^m\{N" 


— 


N') - 




1 

r 


1 




2 


&m^{N" 


' — 


-V)—. 




1 
r 


1 

V 





Sect. 3.] places in orbit. 107 

tion with the third gives rise to the following mode of proceeding. The auxil- 
iary angle C may be determined by the equation 

tanc,— ^. gin i (jv^/ _ j^) 

and we shall have 

tan {i ]\r-\-iJSr'-]- i N" — n) = tan (45° + C) tan I {]V" — JV). 

Two other solutions wholly analogous to this will result from changing the second 
place with the first or third. Since the formulas for - become more complicated 
by the use of this method, it will be better to deduce e and;;, by the method of 
article 80, from two of the equations (I.). The uncertainty in the determination 
of IT by the tangent of the angle i JY -\- ^ JV' -{- i JY'' — 77 must be so decided 
that e may become a positive quantity: for it is manifest that if values 180° dif- 
ferent were taken for 7/, opposite values would result for e. The sign of ]j, how- 
ever, is free from this uncertainty, and the value of p cannot become negative, 
unless the three given points lie in the part of the hyperbola away from the sun, 
a case contrary to the laws of nature which we do not consider in this place. 

That which, after the more difficult substitutions, would arise from the appli- 
cation of the first method in article 78, IT., can be more conveniently obtained in 
the present case in the following manner. Let the first of equations II. be multi- 
plied by cos i {N" — N'), the third by cos h {N' — N\ and let the product of 
the latter be subtracted from the former. Then, lemma I. of article 78 being 
properly applied,* will follow the equation 

* {v-v) ««t^^ " {N"-N') - h {\-\) cotan \ {JST'-JST) 

= - sin I (i\r"_i\r) cos (I i\^-f i N" — n). 

By combining which with the second of equations 11. 77 and - will be found ; thus, 
77 by the formula 



'Putting, that is, in the second formula, A=z\ {N" — N'), B=\N-{-\N"— 77, 0=i {N—N'). 



108 RELATIONS BETWEEN SEVERAL [BoOK I. 



(l _^) cotan ^ {N" — N) — (^- — l) cotan h {N' — N) 

Hence, also, two other wholly analogous formulas are obtained by interchanging 
the second place with the first or third. 

84. 

Since it is possible to determine the whole orbit by two radii vectores given 
in magnitude and position together with one element of the orbit, the time also 
in which the heavenly body moves from one radius vector to another, may be 
determined, if we either neglect the mass of the body, or regard it as known : 
we shall adhere to the foraier case, to which the latter is easily reduced. Hence, 
inversely, it is apparent that two radii vectores given in magnitude and position, 
together with the time in which the heavenly body describes the intermediate 
space, determine the whole orbit. But this problem, to be considered among the 
most important in the theory of the motions of the heavenly bodies, is not so 
easily solved, since the expression of the time in terms of the elements is tran- 
scendental, and, moreover, very complicated. It is so much the more worthy of 
being carefully investigated ; we hope, therefore, it will not be disagreeable to 
the reader, that, besides the solution to be given hereafter, which seems to leave 
nothing further to be desired, we have thought proper to preserve also the one 
of which we have made frequent use before the former suggested itself to me. 
It is always profitable to approach the more difiicult problems in several ways, 
and not to despise the good although preferring the better. We begin with ex- 
plaining this older method. 



We will retain the symbols r, /, N, N', p, e, U with the same meaning, with 
which they have been taken above ; we will denote the difference N' — N by J, 
and the time in which the heavenly body moves from the former place to the 



Sect. 3.] places in orbit. 109 

latter by t Now it is evident that if the approximate value of any one of the 
quantities p, e, U, is known, the two remaining ones can be determined from them, 
and afterwards, by the methods explained in the first section, the time corre- 
sponding to the motion from the first place to the second. If this proves to be 
equal to the given time t, the assumed value o^ p, e, or 77, is the true one, and the 
orbit is found ; bi^t if not, the calculation repeated with another value differing a 
little from the first, will show how great a change in the value of the time corre- 
sponds to a small change in the values of jo, e-, U; whence the correct value will 
be discovered by simple interpolation. And if the calculation is repeated anew 
with this, the resulting time will either agree exactly with that given, or at least 
differ very little from it, so that, by applying new corrections, as perfect an agree- 
ment can be attained as our logarithmic and trigonometrical tables allow. 

The problem, therefore, is reduced to this, — for the case in which the orbit is 
still wholly unknown, to determine an approximate value of any one of the quan- 
tities p, e, 77. We will now give a method by which the value of p is obtained 
with such accuracy that for small values of J it wul require no further correc- 
tion ; and thus the whole orbit will be determined by the first computation with 
all the accuracy the common tables allow. This method, however, can hardly 
ever be used, except for moderate values of J, because the determination of 
an orbit wholly unknown, on account of the very intricate complexity of the 
problem, can only be undertaken with observations not very distant from each 
other, or rather with such as do not involve very considerable heliocentric 
motion. 



Denoting the indefinite or variable radius vector corresponding to the true 
anomaly v — 77 by (), the area of the sector described by the heavenly body in 
the time t will be hf^ ^ d v, this integral being extended from v = iV to v = N', 
and thus, (k being taken in the meaning of article 6), /c^ s} p =A^ ^ dv. . Now it 
is evident from the fomulas developed by Cotes, that ii (fx expresses any 
function whatever of x, the continually approximating value of the integral 
/(fx .dx taken from x =: u to x = u -\- J is given by the formulas 



110 RELATIONS BETWEEN SEVERAL [BotK i. 

l-J((pu-\-(p{u-^ J)) 

i z/ (9) ?( + 3 9> (m -f 1 z/) 4- 3 9 (m + f //) -|- 9 (m + J)), etc. 

It will be sufficient for our purpose to stop at the two first formulas. 
By the first formula we have in our problem, 



if we put 



J ^ ^ ^ ' '' cos 2 w 



— = tan (45° -[- ft>). 



Wherefore, the first approximate value of \^p, which we will put = Sa, will be 

/ Arr' o 

' -^ k < cos 2 0) 

By the second formula we have more exactly 

fq ^ d v = i J (r r -J- r'r -|- 4 ^ i?) , 
denoting by R the radius vector corresponding to the middle anomaly 

Now expressing p by means of r, R,r,N,N-\-k/l,N-\-J according to the for 
mula given in article 82, we find 

4 sin^ ^ ^ sin ^ z/ 

and hence 

cos-^^ ^(}_^ 1 \ 2 sin^ \ A cos oj 2 sin^J^ 

R \T ~T~ "7/ 'p \j {r r cos 2(a) P ' 

"By putting, therefore, 

2 sin^ ^ A ^ (r r' cos 2 w) «. 

cos w ' 

we have 

rf COS 1 J\J { r r' cos 2 ca) 

cosw(l ) 

whence is obtained the second approximate value of y//>, 



Sect. 3.] places m orbit. Ill 

, I 2 a cos^ i A cos^ 2 a) , 



if we put 

fi /cos 4 ^ COS 2 w\2 
2a( — ^ 1 =e. 

\ cos w / 

"Writing, therefore, tt for \/jt?, tt will be determined by the equation 

which properly developed would ascend to the fifth degree. We may put 
71 = q -{- fi, so that q is the approximate value of jt, and fi a very small quantity, 
the square and higher powers of which may be neglected : from which substitu- 
tion proceeds 

or 

■■_ gg'— (gg— «g)(gg— g)" 

and so 

sq'-\-(qq — 8)(aqq-\-4.8q—5ad)q 
{qq—d){q^-^3 8q — 4:ad) 

Now we have in our problem the approximate value of tt, namely, 3 a, which 
being substituted in the preceding formula for q, the corrected value becomes 

243a*s-\-3a{9aa~8) {9aa-}-78) 

" (9aa— 8) (27 aa-{-5 8) 

Putting, therefore, 

-^-—3 ' —r 

27aa~^' (1 — 3/3)a~~'» 

the formula assumes this form, 

ail±r±21f) 
^- 1 + 5^ ' 

and all the operations necessary to the solution of the problem are comprehended 
in these five formulas : — 

I. - = tan (45° -\- (o) 



112 RELATIONS S BETWEEN SEVERAL [BoOK 1. 

m. 



3 A < cos 2 oj 



2 sin'^ \A^{rr' cos 2 w) 
21 a a cos o) 



j^ 2 cos^^ Acqs'' 2 at 

^^- (l_3/3)cos2w ~'^ 

V- i_|_5j3 — Vi^- 

If we are willing to relinquish something of the precision of these fonnulas, it 
will be possible to develop still more simple expressions. Thus, by making cos w 
and cos 2 w = 1, and developing the value of \j p m. n. series proceeding according 
to the powers of//, the fom-th and higher powers being neglected, we have, 

in which /i is to be expressed in parts of the' radius. Wherefore, by making 
we have 



VI. p=j,'{i-ijj + 4^). 



In like manner, by developing \^ p in a series proceeding according to the powers 
of sin J, putting 

r / sin A i // 

we have 

or 

YKL p =/' + ^ sin^ Jsjrr. 

The formulas VII. and VlLl. agree with those which the illustrious Euler has 
given in the Theoria moius phneianmi et comdanm, but formula VI., with that which 
has been introduced in the Recherches et cakiils sur la vraie orbite ellipiique de la 
cmiete de 1769, p. 80. 



Sect. 3.] 



PLACES m ORBIT. 



113 



87. 

The following examples will illustrate the use of the preceding precepts, while 
from them the degree of precision can be estimated. 

I. Let log r = 0.3307640, log / = 0.3222239, z/ = 7° 34' 53".73 = 27293".73, 
t= 21.93391 days. Then is found m = — 33'47".90, whence the further compu- 
tation is as foUows : — 



log J . . 

logrr . . 
C.log3^ . . 
CAogt. . . 
C. log cos 2 0) 



4.4360629 
0.6529879 
5.9728722 
8.6588840 
0.0000840 



1 log y. / QQQ 2 0) 

2 log sin i J 
log-ij . . 
C. log a a 
C. loe; cos w . 



log« . . . . 9.7208910 



0.3264519 
7.0389972 
8.8696662 
0.5582180 
0.0000210 



log|5 6.7933543 

8= 0.0006213757 



log2 . . . . 
2 log cos i J 
2 log cos 2 0) 
Clog (1-3/5) 
2 C. log cos w 



0.3010300 
9.9980976 
9.9998320 
0.0008103 
0.0000420 



1 + / + 21/3 



3.0074471 



log 

log« . . . . 

Clog (1 + 5^) 



0.4781980 
9.7208910 
9.9986528 



logr . . . . 0.2998119 

y = 1.9943982 

21 /5 == 0.0130489 



logy/i? 
log^ 



0.1977418 
0.3954836 



This value of log p differs from the true value by scarcely a single unit in the 
seventh place: formula VI., in this example, gives log j» = 0.3954822; formula 
Vn. gives 0.3954780 ; finally, formula Vm., 0.3954754. 

n. Let log r= 0.4282792, log /= 0.4062033, // = 62° 55' 16".64, if =259.88477 
days. Hence is derived co = — 1° 27'20".14, log a = 9.7482348, /? = 0.04535216, 
y = 1.681127, log s/ p = 0.2198027, logj^ == 0.4396054, which is less than the true 
value by 183 units in the seventh place. For, the true value in this example is 
0.4396237 ; it is found to be, by formula VI, 0.4368730 ; from formula VH. it 

15 



114 EELATIONS BETWEEN SEVERAL [BoOK I. 

results 0.4159824 ; lastly, it is deduced from formula YIH., 0.4051103 : the two 
last values differ so much from the truth that they camiot even be used as ap- 
proximations. 



The exposition of the second method will afford an opportunity for treating 
fully a great many new and elegant relations ; which, as they assume different 
forms in the different kinds of conic sections, it will be proper to treat separately ; 
we will begin with the ELLIPSE. 

Let the eccentric anomalies U, E', and the radii vectores r, r, correspond to 
two places of the true anomaly v, v', (of which v is first in time) ; let also p 
be the semi-parameter, e = sin 9) the eccentricity, a the semi-axis major, t the 
time in which the motion from the first place to the second is completed ; finally 
let us put 
v' — v = 2f, v' ^v = 1F, E' — E=z2g, E'-\-E=2G, acoscp = ^ = b. 

Then, the following equations are easily deduced from the combination of for- 
mulas v., VI., article 8 : — 

[1] ^ sin ^ = sin/, y/ r /, 

[2] bsma = smF.\/rr, 
jt? cosy = (cos I ^' COS I ^;^ (1 -|- e) -[- sin ^ y sin ^ v'. (1 — e)) y/ r r, or 

[3] p cosy = (cos/-j- e cos F) y/ r r, and in the same way, 

[4] JO cos 6^ = (cos J' 4" ^ cos/) y/r/. 
From the combination of the equations 3 and 4 arise, 

[5] cos/. y/r/ = (cosy — e cos G) a, 

[6] cosi^.y/r/= (cos 6^ — e cosy) a. 
From formula III., article 8, we obtain 

[7] / — r = 2 ae siny sin (r, 
r -\-r =^2 a — 2ae cosy cos G =i2a sin^y -\- 2 cos/cosyy/r/j 
whence, 



[8] a 



r -\-r' — 2 cos/cos^'y/r/ 



Sect. 3.] places m oebit. 115 

Let us put 

[9] \/f+/7 ^ 

2 cos/ ' ' 

and then will 

rlQ■^ ^ _ ^ (^ + sin^iff) cos/y/r/ , 

also 

' — sin^ ' 

in which the upper or lower sign must be taken, as sin y is positive or negative. 

Formula XII., article 8, furnishes us the equation 

k t 

— =^E' — esin^' — ^-|-esin^=2^ — 2esin^cos(? 

= 2^ — sin2^-|-2 cos/sin^^^. 

If now we substitute in this equation instead of a its value from 10, and put, for 
,the sake of brevity, 

[11] I 5=^j 

2^cos/2'(r/)* 

we have, after the proper reductions, 

[12] +m = (^+sinn^)^+(^+sinn^)^ f^-:^"^1 . 
in which the upper or lower sign is to be prefixed to m, as sin g is positive or 
negative. 

"When the heliocentric motion is between 180° and 360°, or, more generally, 
when cos/ is negative, the quantity m determined by formula 11 becomes im- 
aginary, and I negative ; in order to avoid which we will adopt in this case, instead 
of the equations 9, 11, the following: — 

1 [9=^] VV+V/_i_2^ 

■- -■ 2 cos/ ' 

[11*] -1 '-^^ ,=M, 

2^(— cos/f (r/f 

whence for 10, 12, we shall obtain these, — 



116 RELATIONS BETWEEN SEVERAL [BoOK I. 

nn:in — 2 (Z— sin^lgr) cos/y /r/ 

[-12.] +Jf^_(i:_sinn^)'+(Z-sin^^^)^-(^lr^^)^ 
in which the doubtful sign is to be determuied in the same manner as before. 



89. 

"We have now two things to accompHsh ; first, to derive the unknown quan- 
tity (/ as conveniently as possible from the transcendental equation 12, since it 
does not admit of a direct solution ; second, to deduce the elements themselves 
from the angle (/ thus found. Before we proceed to these, we will obtain 
a certain transformation, by the help of which the computation of the auxiliary 
quantity I ov L is more expeditiously performed, and also several formulas after- 
wards to be developed are reduced to a more elegant form. 

By introducing the auxihary angle w, to be determined by means of the 
formula 

''4 = tim(45° + <»), 



^ 



we have 

J^-\- sjy = 2 -f (tan (45° + t«) — cotan (46° -f w))^ = 2 + 4 tan^ 2 w; 
whence are obtained 

, sia^l/ I tan2 2a) ^ sin^i/ tan'^ 2 to 

cos/ "T" cos/ ' cos/ cos/ 

90. 

"We will consider, in the first place, the case in which a value of </ not very 

great, is obtained from the solution of the equation 12, so that 

2ff — sm2g 
sin^ g 

may be developed in a series arranged according to the powers of sin J (/. The 

numerator of this expression, which we shall denote by X, becomes 

-\2- sin^ h (/ — -\-^- sin^ ^ y — | sin' ^ (/ — etc. ; 



Sect. 3.] ' places m oebit. 117 

and the denominator, 

8 sin^ i^ — 12 sin^ a ^ + 3 sin^ 2 (/ -\- etc. 
Whence X obtains the form 

I +1 sin^ i^ + II sinn^+ etc. 

But in order to obtain the law of progression of the coefficients, let us differen- 
tiate the equation 

X sin^^ = 2 y — sin 2 ^, 
whence results 

3Xcos^sin^^-f-^i^^.^x^= 2 — 2 cos 2^ = 4:sin^^; 
putting, moreover, 

sin^ h g=^x, 
"We have 

^g = Hmg, 
whence is deduced 

dX 8 — eXcos^ 4 — 3X(1— 2a;) 

da; sin^^ 2x{l — x) ' 

and next, 

{%x — 2xx)~ = i — {^ — %x)X. 
If, therefore, we put 

X= ^{l^ax -^ ^ XX ^ r ci^ -\-d x''^ Qic.) 
we obtain the equation 

^(ax-{-{2§ — a)xx^{^r — 2^)x^-\-{id — ^y)x'^eiQ?} 
=.{^-^4.a)x-\-{9>a — 4.^)xx-\-{^§ — 4.r)x^^{%r — 4.d)x^^Qic. 
which should be identical. Hence we get 

in which the law of progression is obvious. We have, therefore, 

j^_4 I ^-e^ I ^-e-S I 4.6.8.10 3 ■ 4.6.8.10.12 . ■ 

3 -t-3, 5^-1-3. 5. 7^^-1-3. 5. 7, 9 ^ -h 3.5.7.9.11 ^^ e^c. 
This series may be transformed into the following continuous fraction : — 



118 RELATIONS BETWEEN SEVERAL ' [BoOK 1. 



1 I 2 
1 + 577^ 



T 5.8 
1-779^ 



i-9TTr^ 



1 7.10 

-^~1LT3^ 






-^""15.17^ 



1 — etc. 

The law according to which the coefficients 

6 2_ 5^ h£ 

5' ~577' 779' 9^1' ^ ^' 

proceed is obvious; in truth, the n"" term of this series is, when n is even. 



2»i + 1.2rt + 3' 
when n is odd, 

n-\-2.n-\-o 
2n-\-1.2n/-}-3' 

the further development of this subject would be too foreign from our pm'pose. 
If now we put 



1- 


'7.9 


X 








1- 


1. 

■9T 


.4 

IT 


X 


we have 


1 


— 


etc. 






Sect. 3.] PLACES IN OKBIT. 119 

and 

or 

sm^g — ^(2 g — sm2 g) (l—ism^g) 

The numerator of this expression is a quantity of the seventh order, the denomi- 
nator of the third order, and ^, therefore, of the fourth order, if ^ is regarded as 
a quantity of the first order, and x as of the second order. Hence it is inferred 
that this formula is not suited to the exact numerical computation of ^ when ^ 
does not denote a very considerable angle: then the following formulas are 
conveniently used for this purpose, which differ from each other in the changed 
order of the numerators in the fractional coefiicients, and the first of which is 
derived without difficulty from the assumed value of x — 5.* 

[13] ^ 



1+ 


A 


X- 


-u 


X 




X 

m 






1- 


-■h 


X 








1— 


X 



etc., 



^ = 



■sS 



1 — il^— 6¥^ 



1 4.O.. 



iVs^ 



JO X 



1 — etc. 
In the third table annexed to this work are found, for all values of x from 
to 0.3, and for every thousandth, corresponding values of \ computed to 
seven places of decimals. This table shows at first sight the smallness of 1 for 



* The derivation of the latter supposes some less obvious transformations, to be explained on another 
occasion. 



120 RELATIONS BETWEEN SEVERAL [BoOK I. 

moderate values of g ; thus, for example, for E' — ^=10°, or y = 5°, when 
X = 0.00195, is ^ = 0.0000002. It would be superfluous to continue the table fur- 
ther, since to the last term a;=0.3 corresponds g = 66° 25', or U' — U=^ 132° 50'. 
The third column of the table, which contains values of ^ corresponding to nega- 
tive values of x, will be explained further on in its proper place. 

91. 

Equation 12, in which, in the case we are treating, the upper sign must evi- 
dently be adopted, obtains by the introduction of the quantity ^ the form 

Putting, therefore, 
and 



[14] 



t+z+§ — -' 

the proper reductions being made, we have 

[15] A = fc^^ 

If, accordingly, h may properly be regarded as a known quantity, f/ can be de- 
termined from it by means of a cubic equation, and then we shaU have 

[16] x = "^ — l. 

Now, although h involves the quantity |, still unknown, it will be allowable to 
neglect it in the first approximation, and for h to take 

m m 

since ^ is undoubtedly a very small quantity in the case we are discussing. 
Hence ,y and z will be deduced by means of equations 15, 16 ; | will be got 
from X by table m., and with its aid the corrected value of h will be obtained by 
formula 14, with which the same calculation repeated will give corrected values 
of y and x : for the most part these will differ so little from the preceding, that | 



Sect. 3.] places in orbit. 121 

taken again from table III., will not differ from the first value ; otlierAvise it ^vould 
be necessary to repeat the calculation anew until it underwent no further change. 
When the quantity x shall be found, g will be got by the formula sin^ 2 y = x. 

These precepts refer to the first case, in which cos/ is positive \ in the other 
case, where it is negative, we put 



v/(X-:.) = 5 



and 



["*] x^^.=^. 



whence equation 12* properly reduced passes into this, 

[15.] B= (^%YJ - 
Y and H can be determined, accordingly, by this cubic equation, whence again x 
wiU be derived from the equation 

[16*] x = L — -^^. 

In the first approximation 

MM 

will be taken for H; \ will be taken from table m. with the value of x derived 
from H by means of the equations 15*, 16*; hence, by formula 14*, will be had 
the corrected value of H, with which the calculation will be repeated in the same 
manner. Finally, the angle g will be determined from x in the same way as in 
the first case. 

92. 

Although the equations 15, 15*, can have three real roots in certain cases, it 
will, notwithstanding, never be doubtful which should be selected in our problem. 
Since h is evidently a positive quantity, it is readily inferred from the theory 
of equations, that equation 15 has one positive root with two imaginary or two 
negative. Now since 



16 



122 ■ RELATIONS BETWEEN SEVERAL [BoOK 1. 

must necessarily be a positive quantity, it is evident that no uncertainty remains 
here. So far as relates to equation 15*, we observe, in the first place, that L is 
necessarily greater than 1 ; which is easily proved, if the equation given in article 
89 is put under the form 

r- 1 I cos^ 2/1 ^"^^ 2 (a . 

"^ — cos/ ' — cos/' 

Moreover, by substituting, in equation 12*, YsJ [L — x) in the place of 31, we 

have 

Y^\ = {L-x)X, 
and so 

r+i>(i-:.)X>| + A^ + AL^^_l_A^^4_etc.>|, 

and therefore F^ |^. Putting, therefore, Y=l -\- Y', Y' wiU necessarily be a 
positive quantity ; hence also equation 15* passes into this, 

which, it is easily proved from the theory of equations, cannot have several posi- 
tive roots. Hence it is concluded that equation 15* would have only one root 
greater than i,-|- which, the remaining ones being neglected, it will be necessary 
to adopt in our problem. 

93. 

In order to render the solution of equation 15 the most convenient possible 
in cases the most frequent in practice, we append to this work a special table 
(Table II.), which gives for values of h from to 0.6 the corresponding loga- 
rithms computed with great care to seven places of decimals. The argument 
h, from to 0.04, proceeds by single ten thousandths, by which means the 
second differences vanish, so that simple interpolation suffices in this part 
of the table. But since the table, if it were equally extended throughout, 
would be very voluminous, from h = 0.04 to the end it was necessary to proceed 
by single thousandths only ; on which account, it will be necessary in this latter 
part to have regard to second differences, if we wish to avoid errors of some units 



t If in fact we suppose that our problem admits of solution. 



Sect. 3.] places in orbit. , 123 

in the seventh figure. The smaller values, however, of Ji are much the more fre- 
quent in practice. 

The solution of equation 15, when h exceeds the limit of the table, as also 
the solution of 15*, can be performed without difficulty by the indirect method, 
or by other methods sufficiently known. But it will not be foreign to the pur- 
pose to remark, that a small value of g cannot coexist with a negative value of 
cos/, except in an orbit considerably eccentric, as will readily appear from equa- 
tion 20 given below in article 95."}* 

94. 

The treatment of equations 12, 12* explained in articles 91, 92, 93, rests upon 
the supposition that the angle g is not very large, certainly within the limit 66° 25', 
beyond which we do not extend table III. When this supposition is not correct, 
these equations do not require so many artifices ; they can be most securely 
and conveniently solved by trial tvithout a change of form. Securely, since the value 
of the expression 

2 g — sin 2 ^r 
sin* 5- ' 

in which it is evident that 2g is to be expressed in parts of the radius, can, for 
greater values of ^,be computed with perfect accuracy by means of the trigonomet- 
rical tables, which certainly cannot be done as long as ^ is a small angle : con- 
veniently, because heHocentric places distant from each other by so great an interval 
will scarcely ever be used for the determination of an orbit wholly unknown, while 
by means of equation 1 or 3 of article 88, an approximate value of g follows 
with almost no labor, from any knowledge whatever of the orbit: lastly, from an 
approximate value of g, a corrected value will always be derived with few trials, 
satisfying with sufficient precision equation 12 or 12*. For the rest, when two 
given heliocentric places embrace more than one entire revolution, it is necessary 
to remember that just as many revolutions will have been completed by the eccen- 
tric anomaly, so that the angles^' — E, v' — v, either both lie between and 360°, 



t That equation shows, that if cos/ is negative, qo must, at least, be greater 



than 90°— ^r. 



124 RELATIONS BETWEEN SEVERAL [BoOK I 

or both between similar multiples of the wbole circumference, and also / and g 
together, either between and 180°, or between similar multiples of the semicir- 
cumference. If, finally, the orbit should be wholly unknown, and it should not 
appear whether the heavenly body, in passing from the first radius vector to the 
second, had described a part only of a revolution or, in addition, one entire revo- 
lution, or several, our problem would sometimes admit several different solutions : 
however, we do not dwell here on this case, which can rarely occur in practice. 

95. 

"We pass to the second matter, that is, the determination of the elements from 
the angle g when found. The major semiaxis is had here immediately by the 
formulas 10, 10*, instead of which the following can also be used : — 

2 mm cos f\J rr' khtt 



[17] a: 



yysm^g 4 yy rr cos- fsin^g 



[--, K:::-| — 2 MM COS fsj r v' kktt 

The minor semiaxis h = ^ ap is got by means of equation 1, which being 
combined with the preceding, there results 

Now the elliptic sector contained between two radii vectores and the elliptic arc 

is h kt \J p, also the triangle between the same radii vectores and the chord 

^rr' sin 2/: wherefore, the ratio of the sector to the triangle is as^: 1 or Y: 1. 

This remark is of the greatest importance, and elucidates in a beautiful manner 

both the equations 12,12*: for it is apparent from this, that in equation 12 the 

i. a. 1. 5 

parts m, {l-{-xY, X(/-|-.r) , and in equation 12* the parts M, {L — xY,X{L — xy, 

are respectively proportional to the area of the sector (between the radii vectores 
and the elliptic arc), the area of the triangle (between the radii vectores and the 
chord), the area of the segment (between the arc and the chord), because, the 
first area is evidently equal to the sum or difference of the other two, accord- 
ins; as v — V lies between and 180°, or between 180° and 360°. In the case 



Sect. 3.] places in orbit. 125 

where v — v is greater than 360° we must conceive the area of the whole eUipse 
added to the area of the sector and the area of the segment just as many times 
as the motion comprises entire revolutions. 

Moreover, since 5 =a cos 9, from the combination of equations 1, 10, 10*, 
follow 

n A-i sin q tan f 

[19] cosq) = - ,, , ■ 21 V 

[19*] c«s9 = ^Si^. 
whence, by substituting for I, X, their values from article 89, we have 

rc\f\-y sin /"sin q 

[20] coscp = - -^ — , ^, 2.0 • 

i- -• ' 1 — COS/ COS 5^ -|- 2 tan'' 2 to 

This formula is not adapted to the exact computation of the eccentricity 
when the latter is not great : but from it is easily deduced the more suitable 
formula 

[21] tan^i^) 



sinH(/— 5r)+tan2 2. 



to which the following form can likewise be given (by multiplying the numerator 
and denominator by cos^ 2 w) 



[22] tan^ I 9 



^ sin^ \ if—g) + cos^ \ (/- g) sin'' 2 a> 
sinH {f-\-9) + cosH (/— 5") sin2 2 w* 



The angle 9 can always be determined with all accuracy by either formula, using, 
if thought proper, the auxiliary angles of which the tangents are 

tan 2 CO tan 2 oi 

sini(/— 5^)' sin^CZ+^r) 

for the former, or 



tani(/-^)' tani(/+^) 
for the latter. 

The following formula can be used for the determination of the angle G, 
which readily results from the combination of equations 5, 7, and the following 
one not numbered, 

(/ — r)sin5r 



[23] tan G = l^-^;sm^ 

■- -^ {r-\-r)cosg — 2, cos, jsjrr" 

from which, by introducing w,is easily derived 



126 RELATIONS BETWEEN SEVERAL [BoOK I. 

rOA~\ + /^ sin ^ sin 2 ft) 

L J cos''2ajsia-^(/ — 5^) sin^(/-|-^)-[-sin^2 tocos^'' 

The ambiguity here remaining is easily decided by means of equation 7, which 
shows, that G must be taken between and 180°, or between 180° and 360°, 
as the nimierator in these two formulas is positive or negative. 

By combining equation 3 with these, which flow at once from equation II. 
article 8, 

1 1 2e . . . r^ 
r r p '' 

1,1 2 , 2e . J, 

- + -J- = COS / cos I', 

r ' / P P 

the following will be derived without trouble, 

F251 tani^= (/ — y)sin/ 

L -I 2 cos g^rr' — (r'-\-r)cosf' 

from which, the angle la being introduced, results 

r9fin + TP sin /sin 2 to 

•- -^ cos^ 2 tu sin i- (/ — g) sia^ {f-\-g) — sin^2(«cos/* 

The uncertainty here is removed in the same manner as before. — As soon as 
the angles F and Cr shall have been found, we shall have v = F — /, v' = F-\-f, 
whence the position of the perihelion will be known ; also E= G — g, E'^G -\-g. 
Finally the mean motion in the time t will be 

— ^=2g — 2 ecosG^sin^, 
a* 

the agreement of which expressions will serve to confirm the calculation ; also, 

the epoch of the mean anomaly, corresponding to the middle time between the 

two given times, will be G — esinG cos ^, which can be transferred at pleasure 

to any other time. It is somewhat more convenient to compute the mean 

anomalies for the two given times by the formulas E — emiE, E' — e sin E', and 

to make use of their difference for a proof of the calculation, by comparing it with 

S 



Sect. 3.] places in oebit. , 127 

96. 

The equations in the preceding article possess so much neatness, that there 
may seem nothing more to be desired. Nevertheless, we can obtain certain 
other formulas, by which the elements of the orbit are determined much more 
elegantly and conveniently; but the development of these formulas is a little 
more abstruse. 

We resume the following equations from article 8, which, for convenience, we 
distinguish by new numbers : — 

I. smivJ^ = sm^U^{l-\-e) 

n. cos|yi/^=cos^^\/(l — e) 

m. sin^/y/^=sin^^V(l + «) 

We multiply I. by sin h {F-\-g), II. by cos h {F-[-g), whence, the products bemg 
added, we obtain 

or, because 

v/ (1 -[- e) = cos i 9 -j- sin I 9, \/ (1 — e) = cos ^ 9 — sin I 9, 
cos h {f-{-g) \Jl = cos i 9 cos (i #— ia-\-g) — sm i if go& h {F -\- G) . 

In exactly the same way, by multiplying HI. by sin h {F — g), IV. by cos I {F — g), 
the products being added, appears 

cos i {f-\-g) ^^ = cos icp cos{^F— ^G—g) — sin. | 9 cos h {F-\-G). 

The subtraction of the preceding from this equation gives 

cos i {f-j-g) (^-^_y/-l) = 2 COS i 9 sm^ sin ^ {F— G), 

or, by introducing the auxiliary angle w, 

[27] cos h if-^-g) tan 2 to = sin i {F—G) cos h 9 sin^ ^l^. 



128 RELATIONS BETWEEN SEVERAL [BoOK 1. 

By transformations precisely similar, the development of which we leave to the 
skilful reader, are found 

[28] '^^ = <'Osi{F-G)cosi<i,.mg^''^, 

[29] cos h (/ — y) tan 2 to = sin ^ (i^-|- 6^) sin ^ 9 sin^ t/ —,, 

[30] S^^i^==cos}(-P+ff)smJ9sm^^?^. 

When the first members of these four equations are known, ^ (F — G) and 

1 • V«« 7-» 

cosi9)smyy/^ = P 

will be determined from 27 and 29 ; and also, from 29 and 30, in the same manner, 
^(^+6^) and 

the doubt in the determination of the angles ^ {F — G), i {F-\-G), is to be so 
decided that P and Q may have the same sign as siny. Then ^ cp and 

•will be derived from P and Q. From P can be deduced 

EEJr/ 
« = — r^ , 

and also 

unless we prefer to use the former quantity, which must be 

+ V (2 {l-\- sin^ hg) cos/) =+ y/ (— 2 (P — sin^ i^) cos/), 
for a proof of the computation chiefly, in which case a and p are most conven- 
iently determined by the formulas 

7 sinfJr'/ b , 

= — "^ — , a= , » = ocoscp. 

sin ^ ' cos (J) ' -^ ' 

Several of the equations of articles 88 and 95 can be employed for proving the 
calculation, to which we further add the following; : — 



2 tan 2 to / rr' 
cos 2cu 



v/^^^^^"^^^"^^ 



Sect. 3.] 



PLACES IN ORBIT. 



129 



2 tan 2 to I pp 



2 tan 2 0) 



tan (f sin G sin/^ tan cp sin ^sin^. 



cos 2 CO 

Lastly, the mean motion and the epoch of the mean anomaly will be found in the 
same manner as in the preceding article. 



97. 

We will resume the two examples of article 87 for the illustration of the 
method explained in the 88th, and subsequent articles : it is hardly necessary to 
say that the meaning of the auxiliary angle w thus far adhered to is not to be 
confounded with that with which the same symbol was taken in articles 86, 87. 

I In the first example we have /= 3° 47' 26".865, also 

log ^ = 9.9914599, log tan (45° -\-oi) = 9.997864975, w = — 8' 27''.006. 
Hence, by article 89, 



log sin^ i / 
log cos/ . 



7.0389972 
9.9990488 



log tan^ 2 M 
log cos/ . 



5,3832428 
9.9990488 



7.0399484 
= log 0.0010963480 
and thus ^=0.0011205691, | -)-/= 0.8344539, 
log Jet . . . . 9.5766974 



5.3841940 
= log 0.0000242211 

Further we have 



2 loff k t 



C. log 8 cos^/ 



\ogmm 
log(f + 



9.1533948 
9.0205181 
9.0997636 



7.2736765 
9.9214023 



7.3522742 
The approximate value, therefore, of h is 0.00225047, to which in our table II. 
corresponds \ogyy = 0.0021633. We have, accordingly, 

log 7^7 = 7.2715132, or '"^^ 0.001868587, 
yy yy ' 

17 



130 



RELATIONS BETWEEN SEVERAL 



[Book I 



whence, by formula 16, x = 0.000T480179 : wherefore, since | is, by table m., 
wholly insensible, the values found for h, y, x, do not need correction. Now, the 
determination of the elements is as follows : — 

log a; 6.8739120 

logsin^<7 . . . 8.4369560, hg = rW 2".0286, ^ (/+<7) = 3° 27' 45^4611, 
^ {f—g) = 19'41".4039. Wherefore, by the formulas 27, 28, 29, 30, is had 
log tan 2 w . . . 7.6916214^2 Clog cos 2 w . . . 0.0000052 



log cos ^(/+y) 
log cos ^ (/—<?') 



9.9992065 
9.9999929 



log sin ^(/-f-^) 
log sin I (/—^) 



log P sin ^(i^— 6^) 7.6908279^2 
log P cos i {F— G) 8.7810240 



log Q sin i (P4- G) 
log^cosi(P-f 6^) 



8.7810188 
7.7579709 



7.6916143 w 

7.7579761 



HF-G) = 


— 4°38'4r.54 


i,{FJrG) = 


319 21 38 .05 


F= 


314 42 56 .51 


v = 


310 55 29 .64 


v' = 


318 30 23 .37 


G=z 


324 19.59 


E = 


320 52 15 .53 


E'= 


327 8 23.65 



log P = log i? cos ^9 8.7824527 
log ^ = log i^ sin h (f 7.8778355 

Hence ^ cp = T & 0".935 

9 = 14 12 1 .87 
logP 8.7857960 

For proving the calculation. 

^ log 2 cos/. . . . 0.1500394 

Hog(/+^) = log- 8.6357566 

8.7857960 



i log r r . 
log sin/ . 
Clog sin^ 



0.3264939 
8.8202909 
1.2621765 



logb . 
log cos cp 



0.4089613 
9.9865224 



logp 
log a 



0.3954837 
0.4224389 



log sin (p . 
log 206265 



9.3897262 
5.3144251 



log e in seconds 
log sin ^ . . . 
log sin F' . . 



4.7041513 
9.8000767 « 
9.7344714 w 



log e sin ^ 
log e sin F' 



4.5042280 w 
4.4386227 « 



Sect. 3.] places in orbit. 131 

logk ... 3.5500066 esin^ == — 31932'a4 =— 8°52'12'a4 

flog a . . . 0.6336584 gsin^'= — 27455 .08 =— 7 37 35 .08 

2.9163482 Hence the mean anomaly for the 

logt . . . 1.3411160 first place = 329°44'27".67 

4 2574642 ^^^ ^^^ second = 334 45 58 .73 

Difference = 5 1 31 .06 

Therefore, the mean daily motion is 824".7989. The mean motion in the time 

?; is 1809r.07 = 5° r3r.07. 

II. In the other example we have 

/=3r2r38''.32, w= — 2r50".565, J= 0.08635&59, logmm = 9.3530651, 

-r-T-^, or the approximate value of h = 0.2451454 ; 

to this, in table U., corresponds logj/y = 0.1722663, whence is deduced 

"^ = 0.15163477, X = 0.06527818, 

hence from table III. is taken ^ = 0.0002531. Which value being used, the cor- 
rected values become 

h = 0.2450779, logj/ j/ = 0.1722303, "^ = 0.15164737, x = 0.06529078, 

^ = 0.0002532. 
If the calculation should be repeated with this value of | , differing, by a single 
unit only, in the seventh place, from the first; h, logy?/, and x would not suffer 
sensible change, wherefore the value of x already found is the true one, and we 
may proceed from it at once to the determination of the elements. We shall 
not dwell upon this here, as it differs in nothing from the preceding example. 

III. It will not be out of place, to elucidate by an example the other 
case also in which cos/ is negative. Let v' — v = 224° 0' 0", or /= 112° 0' 0'', 
log r = 0.1394892, log /= 0.3978794, ?;= 206.80919 days. Here we find 
to z=: -f 4° 14'43'' 78, L = 1.8942298, log MM= 0.6724333, the first approximate 
value of log^= 0.6467603, whence by the solution of equation 15* is obtained 

¥= 1.591432, and afterwards x = 0.037037, to which, in table HI, corresponds 
g = 0.0000801. Hence are derived the corrected values log ^= 0.6467931, 

Y= 1.5915107, x= 0.0372195, £ = 0.0000809. The calculation being repeated 



132 RELATIONS BETWEEN SEVERAL [BoOK L 

with this value of I, we have x = 0.0372213, which value requires no further cor- 
rection, since \ is not thereby changed. Afterwards is found h g =^ 11° 7'25'^40, 
and hence ia the same manner as in example I. 

^{F—G)=^ 3°33'63".59 log P = log i^ cos i 9 9.9700507 

i(P+^)= 8 26 6.38 log ^ = logi?sini9 . 9.8580552 

Fz= 1159 59.97 \(p= 37°41'34".27 

y = _ 100 .03 9 = 75 23 8 .54 

v'= +123 59 59.97 \ogR 0.0717096 

^ ^= 40-iii:j.7y For proving the calculation. 

E= —17 22 38.01 iog^y/_2cos/ . . 0.0717097 
E'= +27 7 3.59 ^ 

The angle g) in such eccentric orbits is computed a little more exactly by 
formula 19*, which gives in our example 9= 75° 23' 8''.57; likewise the eccen- 
tricity e is determiaed with greater precision by the formula 

g^l_2sin2(45° — ^9), 

than by e ^ sin y ; according to the former, e = 0.96764630. 

By formula 1, moreover, is found log b = 0.6576611, whence \ogp= 0.0595967, 
log a = 1.2557255, and the logarithm of the perihelion distance 

log j^ = \oga{l — e) = \ogb tan (45° — l^>) = 9.7656496. 

It is usual to give the time of passage through the perihelion in place of the 
epoch of the mean anomaly in orbits approaching so nearly the form of the 
parabola ; the intervals between this time and the times corresponding to the 
two given places can be determiaed from the known elements by the method 
given in article 41, of which intervals the difference or sum (accordiag as the 
perihelion lies without or between the two given places), since it must agree with 
the time t, will serve to prove the computation. The numbers of this third ex- 
ample were based upon the assumed elements in the example of articles 38, 43, 
as indeed that very example had furnished our first place : the trifling differences 
of the elements obtained here owe their origin to the limited accuracy of the 
logarithmic and trigonometrical tables. 



Sect. 3.] places in orbit. 133 



The solution of our problem for the ellipse ia the preceding article, might be 
rendered applicable also to the parabola and hyperbola, by considering the parab- 
ola as an ellipse, in which a and l would be infinite quantities, ^ =^ 90°, finally 
E, E', g, and 6^ = ; and in a like manner, the hyperbola as an ellipse, in which a 
would be negative, and h, E, E', g, G, (f, imaginary : we prefer, however, not to 
employ these hypotheses, and to treat the problem for each of the conic sections 
separately. In this way a remarkable analogy wiU readily show itself between 
all three kinds. 

Eetauiing in the PAEABOLA the sjrmbols p, v, v', F,f, r, r', i with the same sig- 
nification with which they had been taken above, we have from the theory of the 
parabolic motion : — 



[1] ^Z.= cosn-^-/) 
[2] sjl, = ^o^h{F^f) 



^ = tan i {F +/) — tan ^ {F—f) -|- \ tan^ | ( J^ +/) _ | tan^ \ {F—f) 
pi 

= (tan i {F-]-f) — tan i {F—f)) (l + tan i {F-\-f) tan i {F—f) + 
i (tan ^ {F-^f) — tan i {F—f)y) 

2 sin/v/ r r' /2 cos fsjrr' .4: sin^ fr r'X 

P \ P ' 3pp /' 

whence 

Further, by the multiplication of the equations 1, 2, is derived 

[4] ^ = cosi^4-cos/ 
and by the addition of the squares, 

[-5] ^4i^^ = l + cos^cos/. 



134 RELATIONS BETWEEN SEVERAL [BoOK I. 

Hence, cos F being eliminated, 

rfil _ 2r/sinY 

\y\ ^ — ^-|_/_2cos/Vr/* 

If, accordingly, we adopt here also the equations 9, 9*, article 88, the first for 
cos/ positive, the second for cos/ negative, we shall have, 

which values being substituted in equation 3, preserving the symbols m, 31, with 
the meaning established by the equations 11, 11*, article 88, there result 

[8] m=l^-\-il^ 

[8*] jf=_x^4-|i:^. 

These equations agree with 12, 12* article 88, if we there put ^= 0. Hence it is 
concluded that, if two heliocentric places which are satisfied by the parabola, are 
treated as if the orbit were elliptic, it must foUow directly from the application 
of the rules of article 19, that a;= 0; and vice versa, it is readily seen that, if 
by these rules we have a; = 0, the orbit must come .out a parabola instead of 
an ellipse, since by equations 1, 16, 17, 19, 20 we should have b = co, a=co, 
(p = 90. After this, the determination of the elements is easily efiected. Instead 
of p, either equation 7 of the present article, or equation 18 of article 95 f might 
be employed : but for F we have from equations 1, 2, of this article 

tan ^ F=z ^ /T / cotan J / ^ sin 2 cy cotan h f, 

if the auxiliary angle is taken with the same meaning as in article 89. 

We further observe just here, that if in equation 3 we substitute instead of 
p its value from 6, we obtain the well-known equation 

kt = }{r-{-r -\-cosf.^ r/){r~\-r— 2 cos f.sjrr'y \/ 2. 



t Whence it is at once evident that y and T express the same ratios in the parabola as in the 
ellipse. See article 95. 



Sect. 3.J places in orbit. 135 



99. 

We retain, in the HYPERBOLA also, the symbols jt?, v, v',f, F, r, r', t with the 
same meaning, but instead of the major semiaxis a, which is here negative, we 
shall write — a ; we shall put the eccentricity e = ^-^ in the same manner as 
above, article 21, etc. The auxiliary quantity there represented by u, we shall 
put for the first place =-, for the second =^ Oc. whence it is readily inferred 
that c is always greater than 1, but that it differs less from one, other things 
being equal, in proportion as the two given places are less distant from each 
other. Of the equations developed in article 21, we transfer here the sixth and 
seventh shghtly changed in form, 



[1] eos|.= K\/T + \/^)\/^^ 

C2] sin|.:.K^^-^^)^^i±l)-« 

[3] cos^.'=^(v/C'. + y/i^)y/(-^-'>" 

[4] ,\nhv'=.h{sJOc-yJ^)s,l^^ 






From these result directly the following : — 
[5] sin i^=^«(C- 1)^/1^ 
[6] ,inf^ha{c-])sj'-^ 

[8] cos/ = (.(C + i)-(.4-i))^^.^ 
Again, by equation X. article 21, we have 



136 RELATIONS BETWEEN SEVERAL [BoOK 1. 

and hence, 

[9] '^=ie{0-l){c-l), 

This equation 10 combined with 8 gives 

/-\- r — (c -f- -) cos/, y/ r/ 

[11] «= '-^ • 

Putting, therefore, in the same manner as in the ellipse 

2 cos/ ' ' ' 

according as cos/ is positive or negative, we have 
[12] « = -^^ ' 

— S (LA-iNc—J-f) cos f.\/r/ 

[12*] «= Y • 

The computation of the quantity / or X is here made with the help of the auxil- 
iary angle w in the same way as in the ellipse. Finally, we have from equation 
XL article 22, (using the hyperbolic logarithms), 

ii^^WCc— i-— - + ^) — logC'c+log- 

= i.(C'+l)(.-l)-21og., 
or, C being eliminated by means of equation 8, 



ht {c — \)cosf.slr/ 


\-h(cc ^ 


«i- 


r^y^^ cc 



21ogc. 
In this equation we substitute for a its value from 12, 12* ; we then introduce 



Sect. 3.] places in orbit. 13 7 

the symbol m or M, with the same meaning that formulas 11, 11% article 88 give 
it ; and finally, for the sake of brevity, we write 



cc 4 log c 

c c o 



from which result the equations 

[13] m={l—zf^{l — zfz, 
[13*] iJf=_(X + .f+(XH-0)^^, 

which involve only one unknown quantity, z, since Z is evidently a function of z 
expressed by the following formula, 

^_ (1 + 2 ^) V/(2^ + ^z) — log (y/ (1 -f g)-[- V^) 

100. 

In solving the equation 13 or 13* we will first consider, by itself, that case in 
which the value of z is not great, so that Z can be expressed by a series proceed- 
ing according to the powers of z and converging rapidly. Now we have 

(i + 2^)v/(04-^^) = ^*4-f^*+l^*-.., 

and so the numerator of ^ is | s^ -f- 1 0^ . . . ; 
and the denominator, 2 0^ -[- 3 s^ . . . , 
whence, 

^ A ^ . . . . 

In order to discover the law of progression, we differentiate the equation 

2(0 + 00f ^=(1-1-20)^/(^4-00) — log (v/(l + 0) + v/0), 
whence results, all the reductions being properly made, 

2(0 + 00)*^-f-3^(l + 20)s/(0 + 00) = 4v/(0 + 00), 
18 



138 RELATIONS BETWEEN SEVERAL [BoOK 1. 

or 

(2. + 2..)^=4-(3 + 6.)Z, 

whence, in the same manner as in article 90, is deduced 

ry , 4.6 I 4.6.8 4.6.8.10 . , 4.6.8.10.12 . 

^=^-375^ + 3-75:7""- 375777^^ + 3.5.7.9.11 " " ^*«- 

It is evident, therefore, that Z depends upon — s in axactly the same manner 

as X does upon x above in the ellipse ; wherefore, if we put 

C also will be determined in the same manner bj — z as |, above, by a?, so that 



we 


have 










[14] 


L: 






or, 


1 + 


t¥3^ 

l+etc. 


c= 


"1 + 


1 + lf^ 





I + tVb^ 
1 + etc. 

In this way the values of t are computed for z to single thousandths, from 2=0 
up to s =: 0.3, which values are given in the third column of table HI. 

101. 

By introducing the quantity I, and putting 

also 

Vt rn mm J 

[15] f+,+^ = !',or 
[15T ^^ = ^, 



Sect. 3.] places m orbit. 139 

equations 13, 13* assume the form, 

[16] ^^^^ = ^. 

[16*] ^L+^f^^H, 

and so, are wholly identical with those at which we arrived in the ellipse (15, 15*, 
article 91). Hence, therefore, so far as ^ or ^ can be considered as known, t/ or 
Y can be deduced, and afterwards we shall have 

[17] ^=1--^, 

[17*] ^ = f?-X. 

From these we gather, that all the operations directed above for the elhpse serve 
equally for the hyperbola, up to the period when t/ or Y shall have been deduced 
from h or J3[; but after that, the quantity 

mm , r ^^ 

yy ' ^^' 

which, in the ellipse, should become positive, and in the parabola, 0, must in the 
hyperbola become negative : the nature of the conic section will be defined by 
this criterion. Our table will give t, from z thus found, hence will arise the cor- 
rected value of h or H, with which the calculation is to be repeated until all 
parts exactly agree. 

After the true value of s is found, c might be derived from it by means of the 
formula 

(? = l-f 2^4-2^(0 + ^^), 

but it is preferable, for subsequent uses, to introduce also the auxiliary angle n, 
to be determined by the equation 

tan2 w== 2\/(s-|-s0); 
hence we have 

(? = tan2 w 4- v/(l + tan2 2 «) = tan (45° + «). 



140 RELATIONS BETWEEN SEVERAL [BoOK 1. 

102. 

Since y must necessarily be positive, as well in the hyperbola as in the ellipse, 
the solution of equation 16 is, here also, free from ambiguity : -j- but with respect 
to equation 16*, we must adopt a method of reasoning somewhat different from 
that employed in the case of the eUipse. It is easily demonstrated, from the the- 
ory of equations, that, for a positive value of 3%, this equation (if indeed it has 
any positive real root) has, with one negative, two positive roots, which will either 
both be equal, that is, equal to 

i y/ 5 _ i = 0.20601, 
or one will be greater, and the other less, than this Hmit. We demonstrate in 
the following manner, that, in our problem (assuming that z is not a large 
quantity, at least not greater than 0.3, that we may not abandon the use of the 
third table) the greater root is always, of necessity, to be taken. K in equation 
13* in place of Jf, is substituted Zy/(X-|-^),we have 

Y^\ = {L^z)Z>{\-^z)Z, or 

T7-»^ 1 ^ I 4. 6 4. 6. 8 o 1 , 

^> 3 -sTs ^ + 0:7^^-3X7:9^ +^*«- 

whence it is readily inferred that, for such small values of z as we here suppose, 

Y must always be > 0.20601. In fact, we find, on making the calculation, that 

z must be equal to 0.79858 in order that {\-\-z)Z may become equal to this 

limit : but we are far from wishing to extend our method to such great values of z. 

103. 

When z acquires a greater value, exceeding the limits of table m., the equa- 
tions 13, 13* are always safely and conveniently solved by trial in their un- 
changed form ; and, in fact, for reasons similar to those which we have explained 

t It will hardly be necessary to remark, that our table II. can be used, in the hyperbola, as well as 
in the ellipse, for the solution of this equation, as long as h does not exceed its limit. 

X The quantity H evidently cannot become negative, unless C > i 5 but to such a value of t, would 
correspond a value oiz greater than 2.684, thus, far exceeding the limits of this method. 



Sect. 3.] places in orbit. 141 

in article 94 for the ellipse. In such a case, it is admissible to suppose the 
elements of the orbit, roughly at least, known : and then an approximate value 
of n is immediately had by the formula 

, o sin f J rr' 

tan 2 w = — rr^ — r,. 

which readily follows from equation 6, article 99. z also will be had from n by 
the formula 



2 cos 2 w cos 2 )^ ' 

and from the approximate value of z, that value will be deduced with a few 
trials which exactly satisfies the equation 13, 13*. These equations can also be 
exhibited in this form, 

(J ^^'-^^^.9(J ^-'-^^ |Sl|-^^P-^°Stan (45° + : 

^=-(^ + 0-^^) +2(^ + ^5^) I t^2s2^. ■ 

and thus, s beiag neglected, the true value of n can be deduced. 



104. 

It remains to determine the elements themselves from z, n, or c. Putting 
ayj [ee — 1) = /?, we shall have from equation 6, article 99, 

[18] /3==^^^^. 

^ -i ^ tan2w 

combining this formula with 12, 12* article 99, we derive, 
[19] ^(..-l) = tanv' = f^, 
[19*] tanv=_?f^, 

whence the eccentricity is conveniently and accurately computed ; a will result 
from (i and \J [ee — 1) by division, and p by multiplication, so that we have. 



142 RELATIONS BETWEEN SEVERAL [BoOK I. 

2 {I — z) cos/, y^r/ 2 m m cos f.\j rr' kktt 

tan'^2n yy tannin 4 y j' rr' cos*/ tan'' 2 w 

— 2(Z-f-z)cos/.v^r/ — 2MMcos/.^rr' kktt 

ta.n^2n TTta.n'2n 4:TYrr^cosytaji^2n* 

sin/, tan/, ^r/ yysmf.taxif.t^rr' /yr'/sia2f\^ 

^ 2{l — z) ~ ¥,i^ ~ \ kt ) 

— sin/.tan/.y/r/ — rFsin/. tan/, y/r/ /r?Vsin2/\2 

~ 2(2+^) ~~ 2J^M ~\ Yt /* 

The third and sixth expressions for p, which are wholly identical with the form- 
ulas 18, 18*, article 95, show that what is there said concerning the meaning 
of the quantities y, Y, holds good also for the hyperbola. 

From the combination of the equations 6, 9, article 99, is derived 

by introducing therefore \\) and w, and by putting (7=: tan (45° -f-iV), we have 
[20] tan2i\^^ '"7^"/" . 

>- -J sm/coszw 

C being hence found, the values of the quantity expressed by u in article 21, will 
be had for both places ; after that, we have by equation III., article 21, 

Q—c 



tan J V 
tan ^ v' 



Gc — 1 



(Cc-l-l)tanii^' 
or, by introducing for C, c, the angles N, n, 

sin (iV — n) 



[21] tan i V 



cos {N-\- n) tan \ \p 



[22] tan^.'=:-^^-mi^. 
>- -■ cos (iv — n) tan ^ i// 

Hence wUl be determined the true anomalies v,v', the difference of which com- 
pared with 2/ will serve at once for proving the calculation. 

Finally, the interval of time from the perihelion to the time corresponding to 
the first place, is readily determined by formula XI., article 22, to be 



4 

& \ 



cos 2 iy^cos 2 n ^^i^- ^^^ tan (45° + «) / ' 



Sect. 3.] ' places in orbit. 143 

and, in the same manner, tlie interval of time from the perihelion to the time cor- 
responding to the second place, 

/ C"°1^7;l::Vf+"' -Vp.Iogtan H^^ + N) tan (46- + „)). 

If, therefore, the first time is put = T — ^ t, and, therefore, the second = T-\- 1 1, 
we have 

[23] 2'=|'(^^-logtan(45° + i^)), 

whence the time of perihelion passage will be known ; finally, 

m ^ = T(^-Htan(45» + «)), 

which equation, if it is thought proper, can be applied to the final proof of the 
calculation. 

105. 

To illustrate these precepts, we will make an example from the two places 
in articles 23, 24, 25, 46, computed for the same hyperbolic elements. Let, 
accordingly, 

v'—v = ^^°12' 0",or/ = 24° 6' 0", log r = 0.0333585, log / = 0.2008641, 
^=51.49788 days. 
Hence is found 

w = 2° 45' 28".47, I = 0.05796039, • 

^^ or the approximate value of ^ = 0.0644371 ; hence, by table 11., 

log 2/ 2/ = 0.0560848, "^ = 0.05047454, = 0.00748585, 

to which in table HI. corresponds t ^ 0.0000032. Hence the corrected value of 
h is 0.06443691, 

logj^y = 0.0560846, ^^'= 0.05047456, 0= 0.00748583, 
°^ yy 

which values require no further correction, because t, is not changed by them. 

The computation of the elements is as follows : — 



144 



RELATIONS BETWEEN SEVERAL 



[Book L 



loo-g 7.8742399 



0.0032389 



log(l+.) 



log2 . . . 



8.9387394 
0.3010300 



log tan 2 n . 


. . 9.2397694 


2n=: 


9°5rir.816 


n = 


4 55 35 .908 



log sin/ .... 9.6110118 

logs/r/ . . . . 0.1171063 

Clog tan 2 w . . 0.7602306 

log/3 0.4883487 

log tan 1// . . . . 9.8862868 

log a . . . . . 0.6020619 

logp 0.3746355 

(tliey should be 0.6020600 and 0.3746356) 



log sin (iV — n) 
C. log cos {J\f-\- n) 
log cot ^ i/> . . 



8.7406274 
0.0112902 
0.4681829 



log tan iy . . . 9.2201005 

hv= 9°25'29';97 

v= 18 50 59.94 

(it should be 18°51'0") 
log 6 0.1010184 

log tan 2 iV^ . . . 9.4621341 
Clog cos 2 w . . 0.0064539 

9.5696064 

number = 0.37119863 
hyp log tan (45° -f-iV) = 0.28591251 



log tan/ 9.6506199 

log ^ tan 2 w .... 8.9387394 
C.log(/ — 0) . . . 1.2969275 

log tan 1// 9.8862868 

^= 37°34'59".77 

(it should be 37'^ 35' 0") 



Clog I sin/ 
log tan 2 to . 
C log cos 2 w 
log sin t// . . 



log tan 2 N . 

2N = 
iV = 
i\^ — w = 

log sin(iV^-|- n] 
Clogcos(iV— 



0.6900182 
8.9848318 
0.0020156 
9.7862685 



n) 



9.4621341 

16° 9' 46^.253 

8 4 53 .127 

3 9 17 .219 

13 29 .035 

. 9.3523527 
. 0.0006587 



log cot it/; . . . . 0.4681829 

log tan iy' .... 9.8211943 

^v'= 33°3r29".93 

v'= 67 2 59.86 

(it should be 67° 3' 0") 

loge 0.1010184 

log tan 2 w . . . . 9.2397694 
Clogcos2iV' . . . 0.0175142 

9.3583020 

number = 0.22819284 
hyplogtan(45°4-w) = 0.17282621 



Difference 



0.08528612 



Difference = 



0.05536663 



Sect. 3.] places in orbit. 145 

log 8.9308783 log 8.7432480 

|log« 0.9030928 flog a 0.9030928 

C.iog^ 1.7644186 C.logy^ 1.7644186 

logT 1.5983897 l^g^ 0.3010300 

T= ' 39.66338 logt 1.7117894 

t= 51.49788 

Therefore, the perihelion passage is 13.91444 days distant from the time 
corresponding to the first place, and 65.41232 days from the time corresponding 
to the second place. Finally, we must attribute to the limited accuracy of the 
tables, the small differences of the elements here obtained, from those, according 
to which, the given places had been computed. 

106. 

In a treatise upon the most remarkable relations pertaining to the motion 
of heavenly bodies in conic sections, we cannot pass over in silence the elegant 
expression of the time by means of the major semiaxis, the sum r<-\- r, and the 
chord joining the two places. This formula appears to have been first discovered, 
for the parabola, by the illustrious Euler, (Miscell. Berolin, T. VII. p. 20,) who 
nevertheless subsequently neglected it, and did not extend it to the ellipse and 
hyperbola : they are mistaken, therefore, who attribute the formula to the illus- 
trious Lambert, although the merit cannot be denied this geometer, of having 
independently obtained this expression when buried in oblivion, and of having 
extended it to the remaining conic sections. Although this subject is treated by 
several geometers, stiU the careful reader will acknowledge that the following 
explanation is not superfluous. We begin with the elliptic motion. 

We observe, in the first place, that the angle 2/ described about the sun 
(article 88, from which we take also the other symbols) may be assumed to be 
less than 360° ; for it is evident that if this angle is increased by 360°, the time 
is increased by one revolution, or 

^^°=a^X 365.25 days. 
19 



146 RELATIONS BETWEEN SEVEEAL [BoOK 1. 

Now, if we denote the chord by ^, we shall evidently have 

()Q = {r cos v' — r cos vf -\- {r sin y' — r sin vf, 

and, therefore, by equations Vm,, IX., article 8, 

Q Q =1 a a {cos U' — cos^)^-f-^<^ cos^^) (sin^' — sin^f 

:=! 4:aa sin^ (/ {sin? G -j- cos^ cp cos^ G) = 4zaa sin^ y (1 — ee cos^ G) . 

We introduce the auxiliary angle h such, that cos h = e cos G ; at the same time, 
that all ambiguity may be removed, we suppose h to be taken between 0° and 
180°, whence sin h will be a positive quantity. Therefore, as y lies between the 
same limits (for if 2 ^ should amount to 360° or more, the motion would attain to, 
or would surpass an entire revolution about the sun), it readily follows from the 
preceding equation that 9 = 2a sin^ sin^, if the chord is considered a positive 
quantity. Since, moreover, we have 

?--|-/==2a(l — ecos^cos(?) =: 2a{l — cos^cos A), 

it is evident that, if we put h — ^ = (J , /« -|-^ = g , we have, 

[1] r + /— ^ = 2a(l— cos(^) = 4«sinH^, 
[2] r -\- r -\-Q^2a{l — cosf) = 4a sin^ ^ « . 

Finally, we have 

kt = a- {1g — 2 esin^ cos G) = a'- {2g — 2 sin^ cos^), 
or 

[3] Jet = a' {t — smE — {d — smd)). 

Therefore, the angles d and e can be determined by equations 1, 2, from 
r -|- /, q, and a ; wherefore, the time t will be determined, from the same equa- 
tions, by equation 3. If it is preferred, this formula can be expressed thus : 

7 7 1/ 2a—(r-\-r')—Q . 2a—(r4-r')—Q 

/ct = a U arc cos ^rJ — ^ — ^ — sm arc cos ~ — - — ^ 

\ 2a 2a 

2a—Cr-[-r')4-Q , • 2a— (r + r') + 0\ 

— arc cos \ '^ ^ ' '^ -4- sm arc cos v -r vry i _ 

2a ' 2a > 

But an uncertainty remains in the determination of the angles (^, e, by their 
cosines, which must be examined more closely. It appears at once, that d 
must lie between — 180° and -j- 180°, and t between 0° and 360° : but thus 



Sect. 3.] places in orbit. 147 

both angles seem to admit of a double, and the resulting time, of a quadruple, 
determination. "We have, however, from equation 5, article 88, 

cos/. \J rr' =^a (cos_^ — cos ^) = 2 a sin I (^ sin ^ e : 
now, sin i 6 is of necessity a positive quantity, whence we conclude, that cos/ 
and sin i d are necessarily affected by the same sign ; and, for this reason, that 
d is to be taken between 0° and 180°, or between — 180° and 0° according as cos/ 
happens to be positive or negative, that is, according as the heliocentric motion 
happens to be less or more than 180°. ' Moreover, it is evident that d must neces- 
sarily be 0°, for 2/= 180°. In this manner d is completely determined. But 
the determination of the angle e continues, of necessity, doubtful, so that two 
values are obtained for the time, of which it is impossible to determine the true 
one, mil ess it is known from some other source. Finally, the reason of this 
phenomenon is readily seen : for it is known that, through two given points, it 
is pos-sible to describe two different ellipses, both of which can have their focus 
in the same given point and, at the same time, the same major semiaxis j* but 
the motion from the first place to the second in these ellipses is manifestly per- 
formed in unequal times. 

107. 

Denoting by x any arc whatever between — 180° and -J- 180°, and by s the 
sine of the arc J ;f , it is known that, 



Jx=^+|.^^« + i.y^^ + |.^^^+etc. 



Moreover, we have 
and thus 



i sin X = s ^ {1 — ss) = s— i s^ —^s^ — ^^^ s^ — etc. 



X-^X=HU' + i4^'-^hl^^^'+hl^,s^^etc. 



* A circle being described from the first place, as a centre, witb tbe radius 2 a — r, and anoiher, 
from the second place, with the radius 2 a — /, it is manifest that the other focus of the ellipse lies in the 
intersection of these circles. Wherefore, since, generally speaking, two intersections are given, two dif- 
ferent ellipses will be produced. 



148 RELATIONS BETWEEN SEVERAL [BoOK I. 

We substitute in this series for s, successively 

3. 

and we multiply tlie results hy a^ ; and thus obtain respectively, the series, 

T^h2 ^s{r-\-/ — QY-{- etc. 

T¥f32 ^(^ + ^'+Q)^4-etc. 

the sums of which we will denote by T, U. Now it is easily seen, since 

2sinH=+^^:+^, 

the upper or lower sign having effect according as 2/ is less or more than 180°, 
that 

the sign being similarly determined. In the same manner, if for £ is taken the 
smaller value, inferior to 180°, we have 

«*(8 — sin e) = U; 
but the other value, which is the complement of the former to 360°, being taken, 
we evidently have 

a^ (e — sin £) = a* 360° — J/: 
Hence, therefore, are obtained two values for the time t, 

UTT t a^360° U+T 
-^,and-^ 1-- 

108. 

If the parabola is regarded as an ellipse, of which the major axis is infinitely 
great, the expression for the time, found in the preceding article, passes into 

Jlii^ + r^ + QfTir + Z-gf): 



Sect. 3.] places in orbit. 149 

but since this derivation of the formula might perhaps seem open to some doubts, 
we will give another not depending upon the ellipse. 
Putting, for the sake of brevity, 

tsiniv = &, tan ^ / = 6', we have r=^p{l-\-&6),r'=ip{l-\- &'&'), 

l — dd , \ — e'e' . 26 . , 2 6' 

— cos v' = , . .,.„ sm V ■=-—^-—, sm v' : 



Hence follow 

r' cos v' — rcosv=ip{&6 — ^'^'\ r' sin v' — r sin v =^p {&' — 6), 
and thus 

Now it is readily seen that ^' — &= — [^^-^ ^ , is a positive quantity : putting, 
therefore, 

V/(l + l(^'-f ^f)=7j, we have Q=p{&' — 6)rj. 
Moreover, 

r -}- r' = i p {2 -{- 66 -\- &' 6') =p (rjrj -^ i {&' — Bf) : 
wherefore, we have 

From the former equation is readily deduced, 

as fj and 6' — & are positive quantities ; but since i {&' — 6) is smaller or greater 
than rj, according as 

rjrj _ 1 (^'_ ^)2 = 1 _|_ ^&'— 2^^/ 



cos -I y COS I- v' 

is positive or negative, we must, evidently, conclude from the latter equation that 



+ \f'±j=^ = V-H6'-6), 



in which the upper or lower sign is to be adopted, according as the angle de- 
scribed about the sun is less than 180°, or more than 180^ 



150 RELATIONS BETWEEN SEVERAL [BoOK I. 

From the equation, wMch in article 98 follows the second equation, we have, 
moreover, 

whence readily follows, 

the upper or lower sign taking effect, as 2/ is less or more than 180°. 

109. 

If, in the hyperbola, we take the symbols a, 0, c, with the same meaning as in 
article 99, we have, from equations Ylll., IX., article 21, 

/ cos y' — rcosv=^ — ^ (e jlC — jAa 

/ sin / — r sin f =: i (c )l C-\- -^) ccsj [ee — 1) ; 

and consequently, 

Let us suppose that y is a quantity determined by the equation 

since this is evidently satisfied by two values, the reciprocals of each other, we 
may adopt the one which is greater than 1. In this manner 

Moreover, 

. + /=|«(.(. + l)(^+^)_4) = ia((. + i)(y + i)-4), 
and thus, 



Sect. 3.] places in orbit. 151 

Putting, therefore, 

we necessarily have 

but in order to decide the question whether J ^- — y/- is equal to -|- 2 w or — 2n, 
it is necessary to inquire whether y is greater or less than c : but it follows readily 
from equation 8, article 99, that the former case occurs when 2/ is less than 
180°, and the latter, when 2/ is more than 180°. Lastly, we have, from the same 
article, 

J=K^+,-)(^-J)-2l0g^=K^^-el)-*G--)-l0g., + l0g^ 

+ 21og(v/(l-[-72n) + w), 

the lower signs belonging to the case of 2/> 180°. Now, log (v/(l -\- mm) -j- m) 
is easily developed into the following series : — 

This is readily obtained from 



dlog (^\J{l-\-mm)-{-m)^= 



Am 



SJ {l-\-mmy 

There follows, therefore, the formula 

2mv/(l + mm) — 21og(v/(14-mm)+m) = 4(i;;i3__i^i^5_j_i^l:|^^7_etc.), 

and, likewise, another precisely similar, if jw is changed to n. Hence, finally, if we 
put 

^=H'-+'-'-?)*-A.J(r + /-9)*+,/„.i(.+/-?)* 



T"S'432 
.1 



[r-\-/ — qf-\-eiG. 



— T^ls 2 • -^ (^ + ^' + 9)* + etc. 



152 RELATIONS BETWEEN SEVERAL PLACES IN ORBIT. [BoOK I. 

we obtain 

which expressions entirely coincide with those given in article 107, if a is there 
changed into — a. 

Finally, these series, as well for the ellipse as the hyperbola, are eminently 
suited to practical use, when a or a possesses a very great value, that is, where the 
conic section resembles very nearly the parabola. In such a case, the methods 
previously discussed (articles 85-105) might be employed for the solution of the 
problem : but as, in our judgment, they do not furnish the brevity of the solution 
given above, we do not dwell upon the further explanation of this method. 



FOURTH SECTION. 

EELATIONS BETWEEN SEVERAL PLACES IN SPACE. 



110. 

The relations to be considered in this section are independent of the nature of 
the orbit, and will rest upon the single assumption, that all points of the orbit lie 
in the same plane with the sun. But we have thought proper to touch here upon 
some of the most simple only, and to reserve others more complicated and special 
for another book. 

The position of the plane of the orbit is fully determined by two places of 
the heavenly body in space, provided these places do not lie in the same straight 
line with the sun. Wherefore, since the place of a point in space can be assigned 
in two ways, especially, two problems present themselves for solution. 

We will, in the first place, suppose the two places to be given by means of 
heliocentric longitudes and latitudes, to be denoted respectively by l, I', {^, (^ : the 
distances from the sun will not enter into the calculation. Then if the longitude 
of the ascending node is denoted by 9, , the inclination of the orbit to the ecliptic 
by i, we shall have, 

tan ^ = tan ^ sin [l — Q), 

tan /3' = tan i sin (X' — U). 

The determination of the unknown quantities Q , tan i, in this place, is referred 

to the problem examined in article 78, 11. We have, therefore, according to the 

first solution, 

tan ^ sin (X — Q,) = tan ^ , 



tan/ cos (X — Q,) 



tan ^' — tan (3 cos Qf — X) 



20 (153) 



154 RELATIONS BETWEEN SEVERAL [BoOK I. 

likewise, accordiag to the third solution, we find Q by equation 

and, somewhat more conveniently, if the angles ^, /5', are given immediately, and 
not by the logarithms of their tangents : but, for determining i, recourse must be 
had to one of the formulas 



tan i = -T- 



tan^ 



sin(A— g^) sin(?/— g^)' 
Finally, the uncertainty in the determination of the angle 

X — Q,or H+H' — Q, 
by its tangent will be decided so that tan^ may become positive or negative, 
according as the motion projected on the echptic is direct or retrograde : this 
uncertainty, therefore, can be removed only in the case where it may be ap- 
parent in what direction the heavenly body has moved in passing from the first 
to the second place ; if this should be unknown, it would certainly be impossi- 
ble to distinguish the ascending from the descending node. 

After the angles 9,,i, are found, the arguments of the latitude it,u', will be 
obtained by the formulas, 

, tana— Q) X y tan(l'—Q,) 

cos I ' cost ' 

which are to be taken in the first or second semicircle, according as the corre- 
sponding latitudes are north or south. To these formulas we add the following, 
one or the other of which can, at pleasure, be used for proving the calculation : — 
cos u=cos§ cos {X — 9,), cos u' = cos /5' cos {X' — 9,), 

sin/3 . / sins' 

smw = -T-^., sm.u = ~-r, 
^ I -' cose ^ ' COS I 



Sect. 4.] places in space. I.5.5 

111. 

Let us suppose, in the second place, the two places to be given by means of 
their distances from three planes, cutting each other at right angles in the sun ; 
let us denote these distances, for the first place, by x, y, z, for the second, by 
x', y', s, and let us suppose the third plane to be the ecliptic itself, also the posi- 
tive poles of the first and second planes to be situated in N, and 90° -j- N. We 
shall thus have by article 53, the two radii vectores being denoted by r, /, 

x^rcos,u cos {N — Q,)-\-rmi.u sin (iV — 9> ) cos i, . 

y = r sin M cos [N — Q> ) cos i — r cos u sin [N — 9,), 

g = r sin % sin i 

a/ = / cos u' cos [N — 9,)-\-r sin u' sin {N — 9, ) cos «', 

1/ z=r' sin u' cos {N — 9, ) cos i — r' cos u' sin (iV — 9, ), 

z' =. r' sin 1/ sin i. 
Hence it follows that ' 

zy — ys =^ rr sin («' — u) sin [N — 9 ) sin i, 
xz' — zx' =^ rr sin {id — u) cos [ISf — 9 ) sin i, 
xy' — yx' = rr sin [u' — u) cos i. 

From the combination of the first formula with the second will be obtained N — 9 
and rr' sin [%{ — u) mii, hence and from the third formula, i and rr' sin {11 — u) 
will be obtained. 

Since the place to which the coordinates x', y', z', correspond, is supposed pos- 
terior in time, 11! must be greater than u : if, moreover, it is known whether the 
angle between the first and second place described about the sun is less or greater 
than two right angles, rr Bm{ii — u)wii and rr'sin(««' — u) must be positive 
quantities in the first case, negative in the second : then, accordingly, N — 9 
is determined without doubt, and at the same time it is settled by the sign of 
the quantity xy — yx, whether the motion is direct or retrograde. On the othei 
hand, if the direction of the motion is known, it will be possible to decide from 
the sign of the quantity xy' — y x, whether u' — u is to be taken less or greater 
than 180°. But if the direction of the motion, and the nature of the angle 



156 EELATIONS BETWEEN SE\T:RAL [Book I 

c"! escribed about the sun are altogether unkno^vn, it is evident that we cannot dis- 
tinguish between the ascending and descending node. 

It is readily perceived that, just as cos i is the cosine of the inclination of 
the plane of the orbit to the third plane, so sin (iV — Q, ) sin i, cos {N — Q> ) sin i, 
are the cosines of the inclinations of the plane of the orbit to the first and second 
j)lanes respectively ; also that r r' sin {%{ — ii) expresses the double area of the tri- 
angle contained between the two radii vectores, and zy' — yz, xz — zx', x y' — yx\ 
the double area of the projections of this triangle upon each of the planes. 

Lastly, it is evident, that any other plane can be the third plane, provided, 
only, that all the dimensions defined by their relations to the ecliptic, are referred 
to the third plane, whatever it may be. 

112. 

Let rr", y", z", be the coordinates of any third place, and v!' its argument of 
the latitude, r" its radius vector. We will denote the quantities / r" sin [u" — u'), 
rr"mi.{v!' — u),rr'wa.[u' — m), which are the double areas of the triangles be- 
tween the second and third radii vectores, the first and third, the first and second, 
respectively, by n, ri, n". Accordingly, we shall have for x", y", z'\ expressions 
similar to those which we have given in the preceding article for x, y, z, and 
x,y', z ; whence, with the assistance of lemma I, article 78, are easily derived the 
following equations : — 

{)=znx — nV -\- n"x", 

= ny — nY -\- n"y", 

=znz — 7i'z' -f- wV. 
Let now the geocentric longitudes of the celestial body corresponding to these 
three places be a, a', a"\ the geocentric latitudes, /^, \V , 1^"; the distances from the 
earth projected on the ecliptic, (^, d', 8"; the corresponding heliocentric longitudes 
of the earth, X, U, II' \ the latitudes, B, B', B" , which we do not put equal to 
0, in order to take account of the parallax, and, if thought proper, to choose 
any other plane, instead of the ecliptic ; lastly, let i>, Z>', D", be the distances of 
the earth from the sun projected upon the echptic. If, then, x, y, z, are expressed 



Sect. 4.] places in space. I57 

by means of L, B, D, «, /?, d, and the coordinates relating to the second and third 
places in a similar manner, the preceding equations will assume the following 
form : — 

[1] = ?z (d~ cos a -|- Z> cos L) — n [d' cos a -\- D' cos L') 

[2] 0=M((ysina4-i>sinX) — ;/(rsina'4-i)'sinX') 

+ n" {d" sin a" + D" sin L"), 
[3] = w ((^ tan (5 + i? tan B) — n{d' tan §' + 1/ tan B') 

+ n" {d" tan (i" + D" tan ^''). 

If a, /i, i>, X, ^, and the analogous quantities for the two remaining places, are 
here regarded as known, and the equations are divided by n', or by if, five un- 
known quantities remain, of which, therefore, it is possible to eliminate two, or to 
determine, in terms of any two, the remaining three. In this manner these three 
equations pave the way to several most important conclusions, of which we will 
proceed to develop those that are especially important. 

113. 

That we may not be too much oppressed with the length of the formulas, we 
will use the following abbreviations. In the first place we denote the quantity 

tan (i sin [a" — a) -(- tan (i' sin (a — a") -\- tan /3" sin {a —a) 
by (0. 1. 2) : if, in this expression, the longitude and latitude corresponding to 
any one of the three hehocentric places of the earth are substituted for the longi- 
tude and latitude corresponding to any geocentric place, we change the number 
answering to the latter in the symbol (0. 1. 2.) for the Roman numeral which 
corresponds to the former. Thus, for example, the symbol (0. 1. 1.) expresses the 
quantity 

tan /3 sin {L' — a') -f tan /?' sin (a — L') -f tan B' mi{a' — a), 
also the symbol (0. 0. 2), the following, 

tan (i sm {a" — L)-\- tan B sin (a —a") -j- tan §" sin {L — a). 
We change the symbol in the same way, if in the first expression any two heho- 



158 RELATIONS BETWEEN SEVERAL [BoOK I. 

centric longitudes and latitudes of the earth whatever, are substituted for two 
geocentric. If two longitudes and latitudes entering into the same expression are 
only interchanged with each other, the corresponding numbers should also be 
interchanged ; but the value is not changed from this cause,' but it only becomes 
negative from being positive, or positive from negative. Thus, for example, we 
have 

(0. 1. 2) = — (0. 2. 1) = (1. 2. 0) = — (1. 0. 2) = (2. 0. 1) = — (2. 1. 0). 
All the quantities, therefore, originating in this way are reduced to the nineteen 
following : — 
(0.1.2) 

(o.i.O), (o.i.i), (o.i.n.), (0.O.2), (0.1.2), (o.n.2), (0.1.2), (1.1.2), (n.1.2), 
(0. 0. 1), (0. 0. II), (0. 1, n.), (1. 0. L), (1. 0. n.), (i. i ii), (2. 0. i), (2. 0. n.), 

(2.1.11), 

to which is to be added the twentieth (0. 1. II.). 

Moreover, it is easily shown, that each of these expressions multiplied by the 
product of the three cosines of the latitudes entering into them, becomes equal 
to the sextuple volume of a pyramid, the vertex of which is in the sun, and the 
base of which is the triangle formed between the three points of the celestial 
sphere which correspond to the places entering into that expression, the radius 
of the sphere being put equal to unity. When, therefore, these three places lie in 
the same great circle, the value of the expression should become equal to ; and 
as this always occurs in three heliocentric places of the earth, when we do not 
take account of the parallaxes and the latitudes arising from the perturbations of 
the earth, that is, when we suppose the earth to be exactly in the plane of the 
ecliptic, so we shall always have, on this assumption, (0. 1. II.) =^ 0, which is, in 
fact, an identical equation if the ecliptic is taken for the third plane. And fur- 
ther, when B, B', B", each, = 0, all those expressions, except the first, become 
much more simple ; every one from the second to the tenth will be made up of 
two parts, but from the eleventh to the twentieth they will consist of only one 
term. 



Sect. 4.] places in space. 159 

114. 

By multiplying equation [1] by sin a' tan B" — sin L" tan ^" , equation [2] 

by cos U' tan 1^" — cos a" tan B" , equation [3] by sin {L" — a"\ and adding the 

products, we get, 

[4] = ;z ((0. 2. n.) d -I- (0. 2. n.) i>) —i{ ((1. 2. n.) 8' + (I. 2. II.) Z>') ; 

and in tbe same manner, or more conveniently by an interchange of the places, 

simply 

[5] = « ((0. 1. 1.) 8 + (0. 1. 1.) X>) -h w" ((2. 1. 1.) 8" + (II. 1. 1.) i)") 
[6] = w' ((1. 0. 0.)^' + (I. 0. 0.)i>') — 1{' ((2. 0. 0.)r -j- (II. 0. 0.) i/'). 

If, therefore, the ratio of the quantities w, n, is given, with the aid of equation 4, 

we can determine 8' from d, or d from d' ; and so likewise of the equations 5, 6. 

From the combination of the equations 4, 5, 6, arises the following, 

P^-, (0. 2. II.) a +(0.2. II.) i) (i.o.o.)y+(i.o.o.)D- (2.i.i.) y +(ii.i.i.) 2y^__ 

L'-l (0.1.L)5-i-(O.l.I.)i> '^ (1.2.II.)5'-|-(I.2.II.)Z)'^ (2.0.O.)5"-f (IL0.O.)Zy'~ "^^ 

by means of which, from two distances of a heavenly body from the earth, the 

third can be determined. But it can be shown that this equation, 7, becomes 

identical, and therefore unfit for the determination of one distance from the other 

two, when 

B=B'=B"=0, 
and 

tan /?' tan /3'' sin (Z — a) sin {L" — L') -^ tan |S" tan (3 sin {L' — a') sin (X — X") 

-f tan (^ tan (^ sin (X" — a") sin (X' — X) = 0. 

The following formula, obtained easily from equations I, 2, 3, is free from this 

inconvenience : — 

[8] (0. 1. 2.) dd'r -f (0. 1. 2) Bd'r + (0. 1. 2) D'dd" + (0. 1. n.) ly'dd' 
-I- (0. 1, n.) B'l/'d + (0. 1. n.) x>x>"d' -f- (o. i. 2) x>x>x + (O. i n.) bd'b" = o. 

By multiplying equation 1 by sin a' tan (3'^ — sin a" tan j3', equation 2 by 
cos a'' tan |3' — cos a' tan j3", equation 3 by sin («'' — a'), and adding the products, 
we get 

[9] Q = n({0.1.2) d -\- {0.1.2) B)— n' (1.1.2) jy^n'' (11.1.2) B"' 



160 RELATIONS BETWEEN SEVERAL PLACES IN SPACE. [BoOK I, 

and in the same manner, 

[10] = ?^(0.O.2.)Z> — «'((0.1.2)(^'-f (O.I.2)2X)-f n"(0.n.2)iX', 
[11] = n (0. 1. 0)D — n' (0. 1. 1.) 1/ + n" ((0. 1. 2) ^ + (0. 1. H.) !>'). 

By means of these equations the distances d, d', d", can be derived from the 
ratio between the quantities n, n, n", when it is known. But this conclusion only 
holds in general, and suffers an exception when (0.1.2)= 0. For it can be shown, 
that in this case nothing follows from the equations 8, 9, 10, except a necessary 
relation between the quantities n, n', n", and indeed the same relation from each 
of the three. Analogous restrictions concerning the equations 4, 5, 6, will readily 
suggest themselves to the reader. 

Finally, all the results here developed, are of no utility when the plane of the 
orbit coincides with the ecliptic. For if /?, §', /i", B, B B" are aU equal to 0, 
equation 3 is identical, and also, therefore, all those which follow. 



SECOND BOOK. 



INVESTIGATION OF THE ORBITS OF HEAVENLY BODIES FROM GEOCENTRIC 

OBSERVATIONS. 



FIEST SECTION. 

DETERMINATION OP AN ORBIT FROM THREE COMPLETE OBSERVATIONS. 

115. 

Seven elements are required for the complete determination of the motion 
of a heavenly body in its orbit, the number of v^hich, however, may be dimin- 
ished by one, if the mass of the heavenly body is either known or neglected ; 
neglecting the mass can scarcely be avoided in the determination of an orbit 
wholly unknown, where all the quantities of the order of the perturbations must 
be omitted, until the masses on which they depend become otherwise known. 
Wherefore, in the present inquiry, the mass of the body being neglected, we re- 
duce the number of the elements to six, and, therefore, it is evident, that as many 
quantities depending on the elements, but independent of each other, are re- 
quired for the determination of the unknown orbit. These quantities are neces- 
sarily the places of the heavenly body observed from the earth ; since each one 
of which furnishes two data, that is, the longitude ajid latitude, or the right ascen- 
sion and dechnation, it will certainly be the most simple to adopt three geocentric 
places which will, in general, be sufficient for determining the six unknown ele- 
ments. This problem is to be regarded as the most important in this work, and, 
for this reason, vrill be treated with the greatest care in this section. 

21 (161) 



J62 DETERmNATION OF AN ORBIT FROM [BoOK II. 

But in the special case, in which the plane of the orhit coincides with the 
ecliptic, and thus both the heliocentric and geocentric latitudes, from their nature, 
vanish, the three vanishing geocentric latitudes cannot any longer be considered 
as three data independent of each other: then, therefore, this problem would 
remain indeterminate, and the three geocentric places might be satisfied by an 
infinite number of orbits. Accordingly, in such a case, four geocentric longitudes 
must, necessarily, be given, in order that the four remaining unknown elements 
(the inclination of the orbit and the longitude of the node being omitted) may be 
determined. But although, from an indiscernible principle, it is not to be ex- 
pected that such a case would ever actually present itself in nature, nevertheless, 
it is easily imagined that the problem, which, in an orbit exactly coinciding with 
the plane of the ecliptic, is absolutely indeterminate, must, on account of the 
limited accuracy of the observations, remain nearly indeterminate in orbits very 
little inclined to the ecliptic, where the very slightest errors of the observations 
are sufficient altogether to confound the determination of the unknown quan- 
tities. Wherefore, in order to examine this case, it will be necessary to select 
six data : for which purpose we will show in section second, how to determine an 
unknown orbit from four observations, of which two are complete, but the other 
two incomplete, the latitudes or declinations being deficient. 

Finally, as all our observations, on account of the imperfection of the instru- 
ments and of the senses, are only approximations to the truth, an orbit based 
only on the six absolutely necessary data may be still liable to considerable 
errors. In order to duninish these as much as possible, and thus to reach the 
greatest precision attainable, no other method will be given except to accumulate 
the greatest number of the most perfect observations, and to adjust the elements, 
not so as to satisfy this or that set of observations with absolute exactness, but 
so as to agree with all in the best possible manner. For which pm^pose, we will 
show in the third section how, according to the principles of the calculus of 
probabilities, such an agreement may be obtained, as will be, if in no one place 
perfect, yet in all the places the strictest possible. 

The determination of orbits in this manner, therefore, so far as the heavenly 
bodies move in them according to the laws of Kepler, will be carried to the 



Sect. 1.] three complete observations. 163 

highest degree of perfection that is desired. Then it will be proper to undertake 
the final correction, in which the perturbations that the other planets cause in the 
motion, will be taken account of: we will indicate briefly in the fourth section, 
how these may be taken account of, so far at least, as it shall appear consistent 
with our plan. 

116. 

Before the determination of any orbit from geocentric observations, if the 
greatest accuracy is deshed, certain reductions must be applied to the latter on 
account of nutation, precession, parallax, and aberration : these small quantities 
may be neglected in the rougher calculation. 

Observations of planets and comets are commonly given in apparent (that 
is, referred to the apparent position of the equator) right ascensions and declina- 
tions. Now as this position is variable on account of nutation and precession, 
and, therefore, different for different observations, it will be expedient, first of all, 
to introduce some fixed plane instead of the variable plane, for which purpose, 
either the equator in its mean position for some epoch, or the ecliptic might be 
selected : it is customary for the most part to use the latter plane, but the former 
is recommended by some peculiar advantages which are not to be despised. 

When, therefore, the plane of the equator is selected, the observations are in 
the first place to be freed from nutation, and after that, the precession being 
applied, they are to be reduced to some arbitrary epoch : this operation agrees 
entirely with that by which, from the observed place of a fixed star, its mean 
place is derived for a given epoch, and consequently does not need explanation 
here. But if it is decided to adopt the plane of the ecliptic, there are two courses 
which may be pursued : namely, either the longitudes and latitudes, by means of 
the mean obliquity, can be deduced from the right ascensions and declinations 
corrected for nutation and precession, whence the longitudes referred to the mean 
equinox will be obtained ; or, the latitudes and longitudes will be computed more 
conveniently from the apparent right ascensions and dechnations, using the appar- 
ent obhquity, and will afterwards be freed from nutation and precession. 

The places of the earth, corresponding to each of the observations, are com- 



164 DETERMDWATION OF AN ORBIT FROM [BoOK 11. 

puted from the solar tables, but they are evidently to be referred to the same 
plane, to which the observations of the heavenly body are referred. For which 
reason the nutation will be neglected in the computation of the longitude of the 
sun ; but afterwards this longitude, the precession being applied, will be reduced 
to the fixed epoch, and increased by 180 degrees; the opposite sign will be given 
to the latitude of the sun, if, indeed, it seems worth while to take account of it : 
thus will be obtained the heliocentric place of the earth, which, if the equator is 
chosen for the fundamental plane, may be changed into right ascension and decli- 
nation by making use of the mean obhquity. 

117. 

The position of the earth, computed in this manner from the tables, is the 
place of the centre of the earth, but the observed place of the heavenly body 
is referred to a point on the surface of the earth : there are three methods of 
remedying this discrepancy. Either the observation can be reduced to the centre 
of the earth, that is, freed from parallax ; or the heliocentric place of the earth 
may be reduced to the place of observation, which is done by applying the 
parallax properly to the place of the sun computed from the tables ; or, finally, 
both positions can be transferred to some third point, which is most conveniently 
taken in the intersection of the visual ray with the plane of the ecliptic ; the 
observation itself then remains unchanged, and we have explained, in article 72, 
the reduction of the place of the earth to this point. The first method cannot be 
applied, except the distance of the heavenly body from the earth be approxi- 
mately, at least, known : but then it is very convenient, especially when the 
observation has been made in the meridian, in which case the declination only is 
affected by parallax. Moreover, it will be better to apply this reduction imme- 
diately to the observed place, before the transformations of the preceding article 
are undertaken. But if the distance from the earth is still whoUy unknown, 
recourse must be had to the second or third method, and the former will be em- 
ployed when the equator is taken for the fundamental plane, but the third will 
have the preference when all the positions are- referred to the ecliptic. 



Sect. 1.] theee coiviplete observations. 165 



118. 

If the distance of a heavenly body from the earth answering to any observa- 
tion is already approximately known, it may be freed from the effect of aberra- 
tion in several ways, depending on the different methods given in article 71. 
Let ^ be the true time of observation ; 6 the interval of time in which light passes 
from the heavenly body to the earth, which results from multiplying 493' into the 
distance ; I the observed place, I the same place reduced to the time ^ -)- ^ by 
means of the diurnal geocentric motion ; V the place I freed from that part of the 
aberration which is common to the planets and fixed stars ; L the true place of 
the earth corresponding to the time t (that is, the tabular place increased by 
20".25) ; lastly, 'X the true place of the earth corresponding to the time t — ^. 
These things being premised, we shall have 

I. I the true place of the heavenly body seen from 'L at the time t — ^. 
n. / the true place of the heavenly body seen from L at the time t. 
ni. V the true place of the heavenly body seen from L at the time t — ^. 
By method L, therefore, the observed place is preserved unchanged, but the fic- 
titious time t — ^ is substituted for the true, the place of the earth being com- 
puted for the former ; method 11., applies the change to the observation alone, but 
it requires, together with the distance, the diurnal motion ; in method III., the 
observation undergoes a correction, not depending on the distance ; the fictitious 
time t — ^ is substituted for the true, but the place of the earth corresponding to 
the true time is retained. Of these methods, the first is much the most conven- 
ient, whenever the distance is known well enough to enable us to compute the 
reduction of the time with sufficient accuracy. But if the distance is whoUy un- 
known, neither of these methods can be immediately appUed : in the first, to be 
sure, the geocentric place of the heavenly body is known, but the time and the 
position of the earth are wanting, both depending on the unknown distance ; in 
the second, on the other hand, the latter are given, and the former is wanting; 
finally, in the third, the geocentric place of the heavenly body and the position 
of the earth are given, but the time to be used with these is wanting. 



166 DETERmNATION OF AN ORBIT FROM [BoOK II. 

"WTiat, therefore, is to be done with our problem, if, in such a case, a sohition 
exact with respect to aberration is required? The simplest course undoubtedly 
is, to determine the orbit neglecting at first the aberration, the effect of which can 
never be important ; the distances will thence be obtained with at least such pre- 
cision that the observations can be freed from aberration by some one of the 
methods just explained, and the determination of the orbit can be repeated with 
greater accuracy. Now, in this case the third method will be far preferable to the 
others : for, in the first method all the computations depending on the position of 
the earth must be commenced again from the very beginning; in the second (which 
in fact is never applicable, unless the number of observations is sufficient to obtain 
from them the diurnal motion), it is necessary to begin anew all the computations 
depending upon the geocentric place of the heavenly body ; in the third, on the 
contrary, (if the first calculation had been already based on geocentric places 
freed from the aberration of the fixed stars) all the preliminary computations 
depending upon the position of the earth and the geocentric place of the heavenly 
body, can be retained unchanged in the new computation. But in this way it 
will even be possible to include the aberration directly in the first calculation, if 
the method used for the determination of the orbit has been so arranged, that 
the values of the distances are obtained before it shall have been necessary to 
introduce into the computation the corrected times. Then the double compu- 
tation on account of the aberration will not be necessary, as will appear more 
clearly in the further treatment of our problem. 

119. 

It would not be difficult, from the connection between the data and unknown 
quantities of our problem, to reduce its statement to six equations, or even to less, 
since one or another of the unknown quantities might, conveniently enough, be 
eliminated : but since this connection is most complicated, these equations would 
become very intractable ; such a separation of the unknown quantities as finally 
to produce an equation containing only one, can, generally speaking, be regarded 



Sect. 1.] three complete observations. 167 

as impossible,* and, therefore, still less will it be possible to obtain a complete 
solution of the problem bj direct processes alone. 

But our problem may at least be reduced, and that too in various ways, to the 
solution of two equations X = 0, F= 0, in which only two unknown quantities 
X, y, remain. It is by no means necessary that x, y, should be two of the ele- 
ments : they may be quantities connected with the elements in any manner 
whatever, if, only, the elements can be conveniently deduced from them when 
found. Moreover, it is evidently not requisite that X, Y, be expressed in explicit 
functions oi x,y : it is sufficient if they are connected with them by a system of 
equations»in such manner that we can proceed from given values of x, y, to the 
corresponding values of X, Y. 

120. 

Since, therefore, the nature of the problem does not allow of a further reduc- 
tion than to two equations, embracing indiscriminately two unknown quantities, 
the principal point will consist, first, in the suitable selection of these unknown 
quantities and arrangemeni of the equations, so that both X and Y may depend 
in the simplest manner upon x, y, and that the elements themselves may follow 
most conveniently from the values of the former when known : and then, it will 
be a subject for careful consideration, how values of the unknown quantities satis- 
fying the equations may be obtained by processes not too laborious. If this should 
be practicable only by blind trials, as it were, very great and indeed almost intol- 
erable labor would be required, such as astronomers who have determined the 
orbits of comets by what is called the indirect method have, nevertheless, often 
undertaken : at any rate, the labor in such a case is very greatly lessened, if, in 
the first trials, rougher calculations suffice until approximate values of the im- 
known quantities are found. But as soon as an approximate determination is 
made, the solution of the problem can be completed by safe and easy methods, 
which, before we proceed further, it will be well to explain in this place. 

* When the observations are so near to each other, that the intervals of the times may be treated as 
infinitely small quantities, a separation of this kind is obtained, and the whole problem is reduced to the 
solution of an algebraic equation of the seventh or eighth degree. 



168 DETERINIINATION Ol' AN ORBIT FROM [BoOK 11. 

The equations X=0, F=i: will be exactly satisfied if for x and // their 
true values are taken ; if, on the contrary, values different from the true ones are 
substituted for x and y, then X and Y will acquire values differing from 0. The 
more nearly x and ?/ approach their true values, the smaller should be the result- 
ing values of X and Y, and when their differences from the true values are very 
small, it will be admissible to assume that the variations in the values of X and Y 
are nearly proportional to the variation of x, if 2/ is not changed, or to the varia- 
tion of y, if a; is not changed. Accordingly, if the true values of x and ?/ are 
denoted by ^, r], the values of X and Y corresponding to the assumption that 
x=^'^ -\-l, i/ = r}-\- fi, will be expressed in the form 

X=al-\-(S^, Y=yl-\-dfj., 
in which the coefficients a, /5, y, d can be regarded as constant^ as long as I and ix 
remain very small. Hence we conclude that, if for three systems of values of 
X, t/, differing but little from the true values, corresponding values of X, Y have 
been determined, it will be possible to obtain from them correct values of x, y so 
far, at least, as the above assumption is admissible. Let us suppose that, 
for ;?• = a, ^ = ^ we have X-= A, Y=:B, 
x = d,y=y X= A Y= B', 

x=d',y=l" X=A' Y=B", 

and we shall have 

A = a{a-l)^(i{h — ri), B = y {^ -I) -^H^ -^l 
A:=.a{a'-l)J^(i{h' — r^),B' = r{a' — l)^-d{h' — ri), 
A'=a{a"-^)Jr(i{h"-ri),B"=.y{d'-l)^d{h"-n). 
From these we obtain, by eliminating a, §, y, d, 

t _ g {A'B' — J!'B') -|- a' {A"B— A B") -]- d' (^ F —A'B) 
^~ A'B' — A!'B-\-A!'B—AB'-\-AB — A'B ' 

_ h{A'B' — A!'B) + h' (A"B— A B') + h" (AB — A'B) 
^~" A'B' — A"B-j-A"B—AB'-\-AB — A'B ' 

or, in a form more convenient for computation, 



^=a- 



(a^— a) (ATB — A B") -^{d' — a){A B — A' B) 
A'B'~A"B-\-A"B—AB'-\-AB — A'B = 

_^ I {h' — b)(A"B—AB')-{-{b" — b)(AB — A'B) 
V—^'T A'B' — A"B-{-A"B—AB'-\-AB — A'B ' 



Sect. 1.] three complete observations. 1G9 

It is evidently admissible, also, to interchange in these formulas the quantities 
a, b, A, B, with a', b', A', B', or with a', V\ A", B". 

The common denominator of all these expressions, which may be put under 
the form {A' — A) [B" — B) — [A' — A) {B' — B), becomes 

{^ad—iiy) {{a' — a) {h"—h) — {a" — a) {h' — b)) : 

whence it appears that a, a, d', b, b', b" must be so taken as not to make 



h"—h V—W 
otherwise, this method would not be applicable, but would furnish, for the values 
of I and 7], fractions of which the numerators and denominators would vanish at 
the same time. It is evident also that, if it should happen that ad — (:?/ = 0, the 
same defect wholly destroys the use of the method, in whatever way a, a', d', 
h, b', b", may be taken. In such a case it would be necessary to assume for the 
values of X the form 

and a similar one for the values of Y, which being done, analysis would supply 
Boiethods, analogous to the preceding, of obtaining from values of X, Y, computed 
for four systems of values of x, y, true values of the latter. But the computation 
in this way would be very troublesome, and, moreover, it can be shown that, in 
such a case, the determination of the orbit does not, from the nature of the ques- 
tion, admit of the requisite precision : as this disadvantage can only be avoided 
by the introduction of new and more suitable observations, we do not here dwell 
upon the subject. 

121. 

When, therefore, the approximate values of the unknown quantities are ob- 
tained, the true values can be derived from them, in the manner just now ex- 
plained, with all the accuracy that is needed. First, that is, the values of X, Y, 
corresponding to the approximate values (a, h) will be computed : if they do not 
vanish for these, the calculation wiU be repeated with two other values [a, b') 
differing but little from the former, and afterwards with a third system [a", b") 

22 



170 DETEEMINATION OF AN ORBIT FROM [BoOK II. 

unless X, Y, have vanislied for the second. Then, the true values ■will be de- 
duced by means of the formulas of the preceding article, so far as the assumption 
on which these formulas are based, does not differ sensibly from the truth. In 
order that we may be better able to judge of which, the calculation of the values 
of X, Y, will be repeated with those corrected values ; if this calculation shows 
that the equations X= 0, Y:= 0, are, still, not satisfied, at least much smaller 
values of X, Y, will result therefrom, than from the three former h3^otheses, and 
therefore, the elements of the orbit resulting from them, will be much more exact 
than those which correspond to the first hypotheses. If we are not satisfied 
with these, it will be best, omitting that hypothesis which produced the greatest 
differences, to combine the other two with a fourth, and thus, by the process of 
the preceding article, to obtain a fifth system of the values of :r, ^ ; in the same 
manner, if it shall appear worth while, we may proceed to a sixth hypothesis, 
and so on, until the equations X=^0, Y=: 0, shall be satisfied as exactly as the 
logarithmic and trigonometrical tables permit. But it will very rarely be neces- 
sary to proceed beyond the fourth system, unless the first hypotheses were very 
far from the truth. 

122. 

As the values of the unknown quantities to be assumed in the second and third 
hypotheses are, to a certain extent, arbitrary, provided, only, they do not differ 
too much from the first hypothesis ; and, moreover, as care is to be taken that the 
ratio [a" — a) : {b" — b) does not tend to an equality with {a — a) : {b' — ^), it is 
customary to put a'=a, b'^=b. A double advantage is derived from this; for, not 
only do the formulas for ^, rj, become a little more simple, but, also, a part of the 
first calculation will remain the same in the second hypothesis, and another part 
in the third. 

Nevertheless, there is a case in which other reasons suggest a departure from 
this custom : for let us suppose X to have the form X' — x, and Y the form 
F' — y, and the functions X', Y', to become such, by the nature of the problem, 
that they are very little affected by small errors in the values of x, y, or that 

\"d^/' Viy/' \d^/' \^) 



Sect. 1.] three complete observations. 171 

may be very small quantities, and it is evident tliat tlie differences between tlie 
values of those functions corresponding to the system x=^, y ^=-% and those 
which result from x = «, y = h, can be referred to a somewhat higher order 
than the differences \ — a,ri — I; but the former values are X' = |, F' := i;, and 
the latter X' =^a^A, Y' = b-\-B, wdience it follows, that a -{-A, h-\-B, are 
much more exact values of x, y, than a, b. If the second hypothesis is based 
upon these, the equations X= 0, F= 0, are very frequently so exactly satisfied, 
that it is not necessary to proceed any further ; but if not so, the third hypoth- 
esis will be formed in the same manner from the second, by making 
it" = a'-\-A!^a^A^A:,b" = b'^B' = h-^B^B', 

whence finally, if it is still not found sufficiently accurate, the fourth will be ob- 
tained according to the precept of article 120. 

123. 

We have supposed in what goes before, that the approximate values of the 
unknown quantities x,y, are already had in some way. Where, indeed, the 
approximate dimensions of the whole orbit are known (deduced perhaps from 
other observations by means of previous calculations, and now to be corrected by 
new ones), that condition can be satisfied without difficulty, whatever meaning we 
may assign to the unknown quantities. On the other hand, it is by no means a 
matter of indifference, in the determination of an orbit still wholly unknown, 
(which is by far the most difficult problem,) what unknown quantities we may 
use ; but they should be judiciously selected in such a way, that the approximate 
values may be derived from the nature of the problem itself Which can be done 
most satisfactorily, when the three observations applied to the investigation of 
an orbit do not embrace too great a heliocentric motion of the heavenly body. 
Observations of this kind, therefore, are always to be used for the first determina- 
tion, which may be corrected afterwards, at pleasure, by means of observations 
more remote from each other. For it is readily perceived that the nearer the ob- 
servations employed are to each other, the more is the calculation affected by their 
unavoidable errors. Hence it is inferred, that the observations for the first de- 



1Y2 DETEEMINATTON OF JlS ORBIT FEOM [BoOK 11. 

termination are not to be picked out at random, but care is to be taken, ^rst, that 
tliey be not too near each other, but then, also, that they be not too distant from 
each other ; for in the first case, the calculation of elements satisfying the obser- 
vations would certainly be most expeditiously performed, but the elements them- 
selves would be entitled to little confidence, and might be so erroneous that they 
could not even be used as an approximation : in the other case, we should aban- 
don the artifices which are to be made use of for an approximate determination 
of the unknown quantities, nor could we thence obtain any other determination, 
except one of the rudest kind, or wholly insufficient, without many more hj-poth- 
eses, or the most tedious trials. But how to form a correct judgment concerning 
these limits of the method is better learned by frequent practice than by rules : 
the examples to be given below will show, that elements possessing great accu- 
racy can be derived from observations of Juno, separated from each other only 22 
days, and embracing a heliocentric motion of 7° 35'; and again, that our method 
can also be applied, with the most perfect success, to observations of Ceres, which 
are 260 days apart, and include a heliocentric motion of 62° 55'; and can give, 
with the use of four hypotheses or, rather, successive approximations, elements 
agreeing excellently well with the observations. 

124. 

We proceed now to the enumeration of the most suitable methods based upon 
the preceding principles, the chief parts of which have, indeed, already been ex- 
plained in the first book, and require here only to be adapted to our purpose. 

The most simple method appears to be, to take for x, ?/, the distances of the 
heavenly body from the earth in the two observations, or rather the logarithms 
of these distances, or the logarithms of the distances projected upon the ecliptic 
or equator. Hence, by article 64, V., will be derived the hehocentric places and 
the distances from the sun pertaining to those places ; hence, again, by article 110, 
the position of the plane of the orbit and the heliocentric longitudes in it ; and 
from these, the radii vectores, and the corresponding times, according to the prob- 
lem treated at length in articles 85-105, all the remaining elements, by which, 
it is evident, these observations will be exactly represented, whatever values may 



Sect. 1.] three complete observations. 173 

have been assigned to x, y. If, accordingly, tlie geocentric place for the time of 
the third observation is computed by means of these elements, its agreement or 
disagreement with the observed place will determine whether the assumed values 
are the true ones, or whether they differ from them ; whence, as a double com- 
parison will be obtained, one difference (in longitude or right ascension) can be 
taken for X, and the other (in latitude or declination) for T. Unless, therefore, 
the values of these differences come out at once = 0, the true values of x, y, may 
be got by the method given in 120 and the following articles. For the rest, it is 
in itself arbitrary from which of the three observations we set out : still, it is bet- 
ter, in general, to choose the first and last, the special case of which we shall speak 
directly, being excepted. 

This method is preferable to most of those to be explained hereafter, on this 
account, that it admits of the most general application. The case must be ex- 
cepted, in which the two extreme observations embrace a heliocentric motion of 
180, or 360, or 640, etc., degrees; for then the position of the plane of the orbit 
cannot be determined, (article 110). It will be equally inconvenient to apply the 
method, when the heliocentric motion between the two extreme observations 
differs very little from 180° or 360°, etc., because an accurate determination of 
the position of the orbit cannot be obtained in this case, or rather, because the 
slightest changes in the assumed values of the unknown quantities would cause 
such great variations in the position of the orbit, and, therefore, in the values of 
-X", Y, that the variations of the latter could no longer be regarded as propor- 
tional to those of the former. But the proper remedy is at hand ; which is, that 
we should not, in such an event, start from the two extreme observations, but from 
the first and middle, or from the middle and last, and, therefore, should take for 
X, F, the differences between calculation and observation in the third or first 
place. But, if both the second place should be distant from the first, and the 
third from the second nearly 180 degrees, the disadvantage could not be removed 
in this way ; but it is better not to make use, in the computation of the elements, 
of observations of this sort, from which, by the nature of the case, it is wholly 
impossible to obtain an accurate determination of the position of the orbit. 

Moreover, this method derives value from the fact, that by it the amount of 



174 DETEKMINATION OF A^ ORBIT FROM [BoOK II. 

the variations which the elements experience, if the middle place changes while 
the extreme places remain fixed, can be estimated without difficulty: in this way, 
therefore, some judgment may be formed as to the degree of precision to be 
attributed to the elements found. 

125. 

We shall derive the second from the preceding method by applying a slight 
change. Starting from the distances in two observations, we shall determine aU 
the elements in the same manner as before; we shall not, however, compute 
from these the geocentric place for the third observation, but will only proceed 
as far as the heliocentric place in the orbit ; on the other hand we wiU obtain the 
same hehocentric place, by means of the problem treated in articles 74, 75, from 
the observed geocentric place and the position of the plane of the orbit; these 
two determinations, different from each other (unless, perchance, the true values 
of X, y, should be the assumed ones), will furnish us X and Y, the difference be- 
tween the two values of the longitude in orbit being taken for X, and the differ- 
ence between the two values of the radius vector, or rather its logarithm, for Y. 
This method is subject to the same cautions we have touched upon in the pre- 
ceding article : another is to be added, namely, that the heliocentric place in orbit 
cannot be deduced from the geocentric place, when the place of the earth happens 
to be in either of the nodes of the orbit ; when that is the case, accordingly, this 
method cannot -be applied. But it wUl also be proper to avoid the use of this 
method in the case where the place of the earth is very near either of the nodes, 
since the assumption that, to small variations of t, y, correspond proportional 
variations of X, Y, would be too much in error, for a reason similar to that which 
we have mentioned in the preceding article. But here, also, may be a remedy 
sought in the interchange of the mean place with one of the extremes, to which 
may correspond a place of the earth more remote from the nodes, except, per- 
chance, the earth, in all three of the observations, should be in the vicinity of the 
nodes. 



Sect. 1.] three cor^iPLETE observations. 175 



126. 

The preceding metliod prepares the way directly for the iliird. In the same 
manner as before, by means of the distances of the heavenly body from the earth 
in the extreme observations, the corresponding longitudes in orbit together with 
the radii vectores may be determined. With the position of the plane of the 
orbit, which this calculation will have furnished, the longitude in orbit and the 
radius vector will be got from the middle observation. The remaining elements 
may be computed from these three heliocentric places, by the problem treated in 
articles 82, 83, which process will be independent of the times of the observa- 
tions. In this way, three mean anomalies and the diurnal motion will be known, 
whence may be computed the intervals of the times between the first and second, 
and between the second and third observations. The differences between these 
and the true intervals will be taken for X and Y. 

This method is less advantageous when the heliocentric motion includes a 
small arc only. For in such a case this determination of the orbit (as we have 
already shown in article 82) depends on quantities of the third order, and does 
not, therefore, admit of sufficient exactness. The slightest changes in the values 
of x,y, might cause very great changes in the elements and, therefore, in the val- 
ues of -X, y, also, nor would it be allowable to suppose the latter proportional to 
the former. But when the three places embrace a considerable heliocentric mo- 
tion, the use of the method will undoubtedly succeed best, unless, indeed, it is 
thrown into confusion by the exceptions explained in the preceding articles, 
which are evidently in this method too, to be taken into consideration. 

127. 

After the three heliocentric places have been obtained in the way we have 
described in the preceding article, we can go forward in the following manner. 
The remaining elements may be determined by the problem treated in articles 
85-105, first, from the first and second places with the corresponding interval of 
time, and, afterwards, in the same manner, from the second and third places and 



176 DETERMINATION OF AN ORBIT FROM [BoOK 11. 

the corresponding interval of time : thus two values will result for each of the 
elements, and from their differences any two may be taken at pleasure for X and 
Y. One advantage, not to be rejected, gives great value to this method ; it is, 
that in the first hypotheses the remaining elements, besides the two which are 
chosen for fixing X and Y, can be entirely neglected, and will finally be deter- 
mined in the last calculation based on the corrected values of x, y^ either from 
the first combination alone, or from the second, or, which is generally preferable, 
from the combination of the first place with the third. The choice of those two 
elements, which is, commonly speaking, arbitrary, furnishes a great variet}^ of 
solutions ; the logarithm of the semi-parameter, together with the logarithm of 
the semi-axis major, may be adopted, for example, or the former with the eccen- 
tricity, or the latter with the same, or the longitude of the perihelion with any 
one of these elements: any one of these four elements might also be combined 
with the eccentric anomaly corresponding to the middle place in either calcula- 
tion, if an elliptical orbit should result, when the formulas 27-30 of article 96, 
will supply the most expeditious computation. But in special cases this choice 
demands some consideration ; thus, for example, in orbits resembling the parabola, 
the semi-axis ma'or or its logarithm would be less suitable, inasmuch as excessive 
variations of these quantities could not be regarded as proportional to changes of 
x,y: in such a case it would be more advantageous to select - . But we give less 
time to these precautions, because the fifth method, to be explained in the follow- 
ing article, is to be preferred, in almost all cases, to the four thus far explained. 

128. 

.Let us denote three radii vectores, obtained in the same manner as in articles 
125, 126, by r, r, r" ; the angular heliocentric motion in orbit from the second to 
the third place by 2/, from the first to the third by If, from the first to the 
second by 2/", so that we have 

next, let 

r'/'sin2/=w, r r" sin 2/' = ?2', r / sin 2/" =: w'' ; 



Sect. 1.] three complete observations. 177 

lastly, let the product of the constant quantity li (article 2) into the intervals of 
the time from the second observation to the third, from the first to the third, and 
from the first to the second be respectively, ^, ^' ^" . The double computation of 
the elements is begun, just as in the preceding article, both from rr f" and ^", 
and from //',/, ^ : but neither computation will be continued to the determina- 
tion of the elements, but v^^ill stop as soon as that quantity has been obtained 
which expresses the ratio of the elliptical sector to the triangle, and which is de- 
noted above (article 91) by y or — Y. Let the value of this quantity be, in the 
first calculation, -»/', in the second, -»]. Accordingly, by means of formula 18, arti- 
cle 95, we shall have for the semi-parameter j>) the two values: — 

But we have, besides, by article 82, a third value, 

4 rr'i" sin/sin/' sin/" 

V— n — n'^ri' ' 

which three values would evidently be identical if true values could have been 
taken in the beginning for x and y. For which reason we should have 

, , „ 4 /9(?"rrV' sin/sin/' sin/" n' dd" 

' Tir{ nn' 2 7]7jrrr cosj cosj cosf 

Unless, therefore, these equations are fully satisfied in the first calculation, we 

can put 

A = logV^> 

Y=^n — n -\-n — ^r—j, — rj, ? — j-, t^- 

' z riri r r r cos J cos/ cos/ 

This method admits of an application equally general with the second ex- 
plained in article 126, but it is a great advantage, that in this fifth method the 
first hypotheses do not require the determination of the elements themselves, but 
stop, as it were, half way. It appears, also, that in this process we find that, as it 
can be foreseen that the new hypothesis will not differ sensibly from the truth, it 
will be sufiicient to determine the elements either from r, /,/", &", alone, or from 
r', r",f, ^, or, which is better, from r, r" f, &'. 

23 



178 DETERMINATION OF AN ORBIT FROM [BoOK II. 

129. 

The five methods thus far explained lead, at once, to as many others which 
differ from the former only in this, that the inclination of the orbit and the lon- 
gitude of the ascending node, instead of the distances from the earth, are taken 
for X and y. The new methods are, then, as follows : — 

I. From X and j/, and the two extreme geocentric places, according to articles 
74, 75, the heliocentric longitudes in orbit and the radii vectores are determmed, 
and, from these and the corresponding times, all the remaining elements ; from 
these, finally, the geocentric place for the time of the middle observation, the 
differences of which from the observed place in longitude and latitude will fur- 
nish X and Y. 

The four remaining methods agree in this, that all three of the heliocentric 
longitudes in orbit and the corresponding radii vectores are computed from the 
position of the plane of the orbit and the geocentric places. But afterwards : — 

II. The remaining elements are determined from the two extreme places only 
and the corresponding times ; with these elements the longitude in orbit and 
radius vector are computed for the time of the middle observation, the differences 
of which quantities from the values before found, that is, deduced from the geo- 
centric place, will produce X and Y: 

in. Or, the remaining dimensions of the orbit are derived from all three 
heliocentric places (articles 82, 83,) into which calculation the times do not enter : 
then the intervals of the times are deduced, which, in an orbit thus found, should 
have elapsed between the first and second observation, and between this last 
and the third, and their differences from the true intervals will furnish us with 
X and Y: 

IV. The remaining elements are computed in two ways, that is, both by the 
combination of the first place with the second, and by the combination of the 
second with the third, the corresponding intervals of the times being used. These 
two systems of elements being compared with each other, any two of the differ- 
ences may be taken for X and Y: 

V. Or lastly, the same double calculation is only continued to the values of 



Sect. 1.] three complete observations. 179 

the quantity denoted by y, in article 91, and then the expressions given in the 
preceding article for X and Y, are adopted. 

In order that the last four methods may be safely used, the places of the earth 
for all three of the observations must not be very near the node of the orbit : on 
the other hand, the use of the first method only requhes, that this condition may 
exist in the two extreme observations, or rather, (since the middle place may be 
substituted for either of the extremes,) that, of the three places of the earth, 
not more than one shall he in the vicinity of the nodes. 

130. 

The ten methods explained from article 124 forwards, rest upon the assump- 
tion that approximate values of the distances of the heavenly body from the 
earth, or of the position of the plane of the orbit, are already known. When 
the problem is, to correct, by means of observations more remote from each other, 
the dimensions of an orbit, the approximate values of which are already, by 
some means, known, as, for instance, by a previous calculation based on other 
observations, this assumption will evidently be liable to no difficulty. But it does 
not as yet appear from this, how the first calculation is to be entered upon when 
all the dimensions of the orbit are still wholly unknown : this case of our problem 
is by far the most important and the most difficult, as may be imagined from 
the analogous problem in the theory of comets, which, as is well known, has 
perplexed geometers for a long time, and has given rise to many fruitless 
attempts. In order that our problem may be considered as correctly solved, that 
is, if the solution be given in accordance with what has been explamed in the 
119th and subsequent articles, it is evidently requisite to satisfy the folio win,: 
conditions : — First, the quantities x, y, are to be chosen in such a manner, that 
we can find approximate values of them from the very nature of the problem, at 
all events, as long as the heliocentric motion of the heavenly body between the 
observations is not too great. Secondly, it is necessary that, for small changes in 
the quantities x, y, there be not too great corresponding changes in the quantities 
to be derived from them, lest the errors accidentally introduced in the assumed 
values of the former, prevent the latter from being considered as approximate. 



180 DETERMINATION OF AN ORBIT FROM [BoOK II. 

Thirdly and lastly, we require that the processes by which we pass from the quan- 
tities X, y, to X, Y, successively, be not too complicated. 

These conditions will furnish the criterion by which to judge of the excellence 
of any method : this will show itself more plainly by frequent applications. The 
method which we are now prepared to explain, and which, in a measure, is to be 
regarded as the most important part of this work, satisfies these conditions so that 
it seems to leave nothing further to be desired. Before entering upon the ex- 
planation of this in the form most suited to practice, we will premise certain pre- 
liminary considerations, and we will illustrate and open, as it were, the way to it, 
which might, perhaps, otherwise, seem more obscure and less obvious. 

131. 

It is shown in article 114, that if the ratio between the quantities denoted 
there, and in article 128 by w, w', rl', were known, the distances of the heavenly 
body from the earth could be determined by means of very simple formulas. 
Now, therefore, if 



should be taken for ar, y^ 



(the symbols ^, ^\ ^", being taken in the same signification as in article 128) im- 
mediately present themselves as approximate values of these quantities in that 
case where the heliocentric motion between the observations is not very great : 
hence, accordingly, seems to flow an obvious solution of our problem, if two dis- 
tances from the earth are obtained from x, y, and after that we proceed agreeably 
to some one of the five methods of articles 124-128. In fact, the symbols ?], if 
being also taken with the meaning of article 128, and, analogously, the quotient 
arising from the division of the sector contained between the two radii vectores 
by the area of the triangle between the same being denoted by rf, we shall have, 

Hl — L i. !^ — ^ !L 



Sect. 1.] three complete observations. 181 

and it readily appears, tliat if n, n', w", are regarded as small quantities of the first 
order, i^ — 1> v' — ^> V" — ^ ^^^> generally speaking, quantities of the second 
order, and, therefore, 

6" 6" 
the approximate values of x,t/, differ from the true ones only by quantities 
of the second order. Nevertheless, upon a nearer examination of the sub- 
ject, this method is found to be wholly unsuitable ; the reason of this we 
will explain in a few words. It is readily perceived that the quantity (0. 1. 2), 
by which the distances in the formulas 9, 10, 11, of article 114 have been multi- 
plied, is at least of the third order, while, for example, in equation 9 the quan- 
tities (0. 1. 2), (1. 1. 2), (II. 1. 2), are, on the contrary, of the first order; hence, 
it readily follows, that an error of the second order in the values of the quanti- 
ties -., "-r produces an error of the order zero in the values of the distances. 
Wherefore, according to the common mode of speaking, the distances would be 
affected by a finite error even when the intervals of the times were infinitely 
small, and consequently it would not be admissible to consider either these dis- 
tances or the remaining quantities to be derived from them even as approximate ; 
and the method would be opposed to the second condition of the preceding 
article. 

132. 

Putting, for^the sake of brevity, 
(0.1.2) = «, (O.L2)ir = — J, (0.O.2)Z) = + e, (0.n.2)i)'' = + c?, 
so that the equation 10, article 114, may become 

US' ^=h -\-c —r A- d —r, 

the coefficients c and d wHl, indeed, be of the first order, but it can be easily 
shown that the difference c — d is to be referred to the second order. Then it 
follows, that the value of the quantity 

cn-\-dn'' 
n-\-n" 



182 DETERMINATION OF AN ORBIT FROM [BoOK IL 

resulting from tlie approximate assumption that n : i{' = ^ : ^" is affected by an 
error of the fourth order only, and even of the fifth only when the middle is dis- 
tant from the extreme observations by equal intervals. For this error is 

where the denominator is of the second order, and one factor of the numerator 
&^" {d — c) of the fourth, the other r(' — ri of the second, or, in that special case, 
of the third order. The former equation, therefore, being exhibited in this form, 

r,/ , I c n -\- d v!' n -\- n" 

ad =^b-{-- \ „ .—h—, 

' n-\-n n ' 

it is evident that the defect of the method explained in the preceding article does 
not arise from the fact that the quantities w, i{' have been assumed proportional to 
^, ^", but that, in addition to this, n' was put propoii;ional to ^'. For, indeed, in this 
way, instead of the factor ^~^" , the less exact value "^ = 1 is introduced, 
from which the true value 



•^ 2 rjt/'rrV cosf cos f cos f" 

differs by a quantity of the second order, (article 128). 

133. 

Since the cosines of the angles/,/',/", as also the quantities rj, if differ from 
unity by a difference of the second order, it is evident, that if instead of 

n' 

the approximate value 

14- "^ 

is introduced, an error of the fourth order is committed. If, accordingly, in place 
of the equation, article 114, the following is introduced, 

an error of the second order will show itself in the value of the distance d' when 



Sect. 1.] three complete observations. 183 

the extreme observations are equidistant from the middle ; or, of the first order in 
other cases. But this new form of that equation is not suited to the determina- 
tion of d', because it involves the quantities r, r', r", still unknown. 

Now, generally speaking, the quantities ^ , ■;7 , differ from unity by a quantity 
of the first order, and in the same manner also the product ^: it is readily 
perceived that in the special case frequently mentioned, this product differs 
from unity by a quantity of the second order only. And even when the orbit 
of the ellipse is slightly eccentric, so that the eccentricity may be regarded as a 
quantity of the first order, the difierence of ^ can be referred to an order one 
degree higher. It is manifest, therefore, that this error remains of the same order 
as before if, in our equation, ^^^^, is substituted for g^, whence is obtained the 
following form, 

In fact, this equation still contains the unknown quantity /, which, it is evident 
nevertheless, can be eliminated, since it depends only on d' and known quantities. 
K now the equation should be afterwards properly arranged, it would ascend to 
the eighth degree. 

134. 

From the preceding it wiU be understood why, in our method, we are about 
to take for x, y, respectively, the quantities 

i"=P, and 2 (^"- !)/»=«. 

For, in the first place, it is evident that if P and Q are regarded as known quanti- 
ties, 8' can be determined from them by means of the equation 

and afterwards d,d'', by equations 4, 6, article 114, since we have 

In the second place, it is manifest that y , ^ ^" are, in the first hypothesis, the 



184 DETERMINATION OF AN ORBIT FROM [BoOK 11. 

obvious approximate values of the quantities P, Q, of which the true values are 
precisely 

6 if" rr"tj7j" cos f cos f cos f" ' 
from vi^hich hypothesis will result errors of the first order in the determination of 
d', and therefore of d, d", or of the second order in the special case several times 
mentioned. Although we may rely with safety upon these conclusions, generally 
speaking, yet in a particular case they can lose their force, as when the quantity 
(0. 1. 2), which in general is of the third order, happens to be equal to zero, or so 
small that it must be referred to a higher order. This occurs when the geocentric 
path in the celestial sphere has a point of contrary flexure near the middle place. 
Lastly, it appears to be required, for the use of our method, that the heliocentric 
motion between the three observations be not too great : but this restriction, by 
the nature of the very complicated problem, cannot be avoided in any way; 
neither is it to be regarded as a disadvantage, since it will always be desired to 
begin at the earliest possible moment the first determination of the unknown 
orbit of a new heavenly body. Besides, the restriction itself can be taken in a 
sufficiently broad sense, as the example to be given below will show. 

135. 

The preceding discussions have been introduced, in order that the principles 
on which our method rests, and its true force, as it were, may be more clearly 
seen : the practical treatment, however, will present the method in an entirely 
different form which, after very numerous applications, we can recommend as 
the most convenient of many tried by us. Since in determining an unknown 
orbit from three observations the whole subject may always be reduced to 
certain hypotheses, or rather successive approximations, it will be regarded as a 
great advantage to have succeeded in so arranging the calculation, as, at the 
beginning, to separate from these hypotheses as many as possible of the compu- 
tations which depend, not on P and Q, but only on a combination of the known 
quantities. Then, evidently, these preliminary processes, common to each hypoth- 
esis, can be gone through once for all, and the hypotheses themselves are reduced 



Sect. 1.] three complete observations. 185 

to the fewest possible details. It will be of equally great importance, if it 
should not be necessary to proceed in every hypothesis as far as the elements, 
but if their computation might be reserved for the last hypothesis. In both 
these respects, our method, which we are now about to explain, seems to leave 
nothing to be desired. 

136. 

We are, in the first place, to connect by great circles three heliocentric places 
of the earth in the celestial sphere. A, A', A" (figure 4), with three geocentric 
places of the heavenly body, B, B', B", and then to compute the positions of these 
great circles with respect to the ecliptic (if we adopt the ecliptic as the funda- 
mental plane), and the places of the points B, B' , B", in these circles. 

Let a, a\ a" be three geocentric longitudes of the heavenly body, /5, ^', /5", lat- 
itudes ; /, t, I", heliocentric longitudes of the earth, the latitudes of which we put 
equal to zero, (articles 117, 72). Let, moreover, /, y', y" be the incHnations to the 
echptic of the great circles drawn from A, A!., A!.', to B, B', B", respectively ; and, 
in order to follow a fixed rule in the determination of these inclinations, we shall 
always measure them from that part of the ecliptic which lies in the direction 
of the order of the signs from the points A, A', A", so that their magnitudes will 
be counted from to 360°, or, which amounts to the same thing, from to 180° 
north, and from to — 180° south. We denote the arcs AB, AB', A'B", which 
may always be taken between and 180°, by d,d', d". Thus we have for the de- 
termination of y and d the formulas, 



n -I J tan 

[1] tany=-. 



sm(« — I) 



[2] i^nd=.'^^^^^^^^. 
To which, if desirable for confirming the calculation, can be added the following, 
sin(^ = ^^, cos(5' = cos/3 cos(a — T). 

sin 7' ' ^ '' 

We have, evidently, entirely analogous formulas for determining y', d\ y", d". Now, 
if at the same time /5 = 0, cf — ^= or 180°, that is, if the heavenly body should 

24 



186 DETERMINATION OF AN ORBIT FROM [BoOK II. 

be in opposition or conjunction and in the ecliptic at the same time, y would be 
indeterminate. But we assume that this is not the case in either of the three 
observations. 

If the equator is adopted as the fundamental plane, instead of the ecliptic, 
then, for determining the positions of the three great circles with respect to the 
equator, will be required the right ascensions of their intersections with the equa- 
tor, besides the inclinations ; and it will be necessary to compute, in addition to 
the distances of the points B, B', B", from these intersections, the distances of the 
points A, A, A" also from the same intersections. Since these depend on the 
problem discussed in article 110, we do not stop here to obtain the formulas. 

137. 

The second step will be the determination of the positions of these three great 
circles relatively to each other, which depend on their inclinations and the places 
of their mutual intersections. If we wish to bring these to depend upon clear 
and general conceptions, without ambiguity, so as not to be obliged to use 
special figures for different individual cases, it will be necessary to premise some 
preliminary explanations. Firstly, in every great circle two opposite directions 
are to be distinguished in some way, which will be done if we regard one of them 
as direct or positive, and the other as retrograde or negative. This being wholly 
arbitrary in itself, we shall always, for the sake of establishing a uniform rule, con- 
sider the dhections from A, A', A" towards B, B', B" as positive \ thus, for example, 
if the intersection of the first circle with the second is represented by a positive 
distance from the point A, it will be understood that it is to be taken from A 
towards B (as D" in our figure) ; but if it should be negative, then the distance 
is to be taken on the other side of A. And secondly, the two hemispheres, into 
which every great circle divides the whole sphere, are to be distinguished by suit- 
able denominations; accordingly, we shall call that the S2<^e;w hemisphere, which, 
to one walking on the inner surface of the sphere, in the positive direction along 
the great circle, is on the right hand ; the other, the inferior. The superior hemi- 
sphere will be analogous to the northern hemisphere in regard to the ecliptic or 
equator, the inferior to the southern. 



Sect. 1.] three complete observations. 187 

These definitions being correctly understood, it will be possible conveniently 
to distinguish both intersections of the two great circles from each other. In fact, 
in one the first circle- tends from the inferior to the superior hemisphere of the 
second, or, which is the same thing, the second from the superior to the inferior 
hemisphere of the first ; in the other intersection the opposite takes place. 

It is, indeed, wholly arbitrary in itself which intersections we shall select for 
our problem ; but, that we may proceed here also according to an invariable rule, 
we shall always adopt these (Z>, D\D", figure 4) where the third circle A'B" passes 
into the superior hemisphere of the second AB\ the third into that of the first 
AB, and the second into that of the first, respectively. The places of these inter- 
sections will be determined by their distances from the points A and A', A and 
A', A and A, which we shall simply denote by AD, A'D, AD', A'D\ AD", AD". 

Which being premised, the mutual inclinations of the circles will be the angles 
which are contained, at the points of intersection D, D', D", between those parts 
of the circles cutting each other that lie in the positive direction ; we shall 
denote these inclinations, taken always between and 180°, by e, &', h". The de- 
termination of these nine unknown quantities from those that are known, evi- 
dently rests upon the problem discussed by us in article 55. We have, conse- 
quently, the following equations : — 

[3] sin ^ £ sin ^ {AD -\- A'D) = sin i {I" — T) sin i {y" + y'\ 
[4] sin J £ cos ^ {AD + A'D) = cos i {f — I') sin h {f — 7'), 
[5] cos I £ sin I {AD — A'D) = sin h {l" — I') cos h {/' + /), 
[6] cos i £ cos ^ {A'D — A'D) = cos ^ {f — I') cos i {/' — /). 
i {AD-\-A"D) and sin ^ £ are made known by equations 3 and 4, i {AD — A'D) 
and cos I- e by the remaining two ; hence A'D, A'D and £. The ambiguity in the 
determination of the arcs i {A'D -\- A'D), i {A'D — A'D), by means of the tan- 
gents, is removed by the condition that sin i e, cos ^ £, must be positive, and the 
agreement between sin ^ £, cos -k e, will serve to verify the whole calculation. 

The determination of the quantities AD', A'D', a', AD", AID', t" is efiected in 
precisely the same manner, and it will not be worth while to transcribe here the 
eight equations used in this calculation, since, in fact, they readily appear if we 
change 



188 



DETERMINATION OF AN ORBIT FROM 



[Book H. 



AD 


A'D 


£ 


r—t 


y" 


/ 


forJLTr 


A'D' 


e' 


i"—i 


f 


Y 


or for AD" 


AD" 


f." 


I' —I 


7' 


7 



respectively. 

A new verification of the whole calculation thus far can be obtained from the 
mutual relation between the sides and angles of the spherical triangle formed by 
joining the three points D, D', D", from which result the equations, true in gen- 
eral, whatever may be the positions of these points, 

%m{A]y — AD') _ sm {A'D — A'D') _ sin (A"D — A"D') 
sin £ sin a' sin n" 

Finally, if the equator is selected for the fundamental plane instead of the eclip- 
tic, the computation undergoes no change, except that it is necessary to sub- 
stitute for the heliocentric places of the earth A, A, A" those points of the equa- 
tor where it is cut by the circles AB, AB', A'B" ; consequently, the right ascen- 
sions of these intersections are to be taken instead of I, X, V, and also instead of 
AD, the distance of the point D from the second intersection, etc. 



138. 

The tliird %iQ.^ consists in this, that the two extreme geocentric places of the 
heavenly body, that is, the points B, B", are to be joined by a great circle, and 
the intersection of this with the great circle AB' is to be determined. Let B^ be 
this intersection, and d' — a its distance from the point A ; let a* be its longitude, 
and /:J=== its latitude. We have, consequently, for the reason that B, B^-, B" lie in 
the same great circle, the well-known equation, 

= tan /5 sin {a" — a=-=) — tan /5'== sin {a" — «) + tan /5" sin (a* — «), 
which, by the substitution of tan y' sin (a* — /') for tan /5% takes the following 
form : — 

= cos (a* — t) (tan /? sin {a" — t) — tan ^" sin (« — f )) 
— sin («=== — /') (tan /5 cos [a" — 1)-\- tan / sin {a" — a) — tan /5" cos (a — T)). 
Wherefore, since tan («■•= — t) = cos y' tan {d' — o) we shall have, 



tan(d' — ff) = 



tan ^ sin (a" — l ') — tan ^' sin (cc — I') 



cos / (tan § cos (a" — V) — tan ^' cos (a — 0) + ^i° 7' ^^^ ("" — ^) ' 



Sect. 1.] three complete observations. 189 

Thence are derived the following formulas, better suited to numerical calculations. 
Putting, 

[7] tan (i sin (a" — l') — tan /5'' sin {a — /') = JSf, 

[8] tan ^ cos {a'' — V) — tan (S" cos (« — I') = Tsuit, 

[9] sm{a" — a) = Tcost, 
we shall have (article 14, IT.) 

[10] UniS'-a) = ^^^y 

The uncertainty in the determination of the arc {d' — a) by means of the 
tangent arises from the fact that the great circles A'B', BB", cut each other in 
tivo points ) we shall always adopt for B^ the intersection nearest the point B' , so 
that o may always fall between the limits of — 90° and -\- 90°, by which means 
the uncertainty is removed. 

For the most part, then, the value of the arc a (which depends upon the 
curvature of the geocentric motion) will be quite a small quantity, and even, gen- 
erally speaking, of the second order, if the intervals of the times are regarded 
as of the first order. 

It wiU readily appear, from the remark in the preceding article, what are the 
modifications to be applied to the computation, if the equator should be chosen 
as the fundamental plane instead of the ecliptic. It is, moreover, manifest that 
the place of the point B^- will remain indeterminate, if the circles ^i>",J.'^" 
should be wholly coincident ; this case, in which the four points J.', B, B\ B" lie in 
the same great circle, we exclude from our investigation. It is proper in the 
selection of observations to avoid that case, also, where the locus of these four 
points differs but little from a great circle ; for then the place of the point B^, 
which is of great importance in the subsequent operations, would be too much 
affected by the slightest errors of observation, and could not be determined with 
the requisite precision. In the same manner the point B^, evidently, remains 
indeterminate when the points B, B" coincide,-]- in which case the position of the 



t Or when they are opposite to each other ; but we do not speak of this case, because our method is 
not extended to observations embracing so great an interval. 



190 DETERMINATION OF AN ORBIT FROM [BoOK II. 

circle BB" itself would become indeterminate. Wherefore we exclude this case, 
also; just as, for reasons sunilar to the preceding, those observations will be 
avoided in which the first and last geocentric places fall in points of the sphere 
near to each other. 

139. 

Let C, C, C", be three heliocentric places of the heavenly body in the celestial 
sphere, which will be (article 64, III.) in the great circles AB, AB', A'B", respec- 
tively, and, indeed, between A and B, A and B', A' and B" ; moreover, the points 
C, C, G" will lie in the same great chcle, that is, in the circle which the plane 
of the orbit projects on the celestial sphere. 

We will denote by r, r', r" , three distances of the heavenly body from the sun ; 
by 9, ^, q", its distances from the earth ; by R, R, R", the distances of the earth 
from the sun. Moreover, we put the arcs CO", CO", CO' equal to 2/, 2f, 2f", 
respectively, and 

rV sin 2/ = n, rr" sin 2/ = ?/, r r sin 2/' = ii'. 
Consequently we have 

f=f-\.f", AO-]-CB = d, AO'-[-0'B' = d', A'0"-^0"B" = d"; 
also, 

sinS sin^C sin (75 



r Q R 

sin 8' sin A' G' sin O'B' 

~?~~ 7 ~ R' 

sin S' _ sin J!' G" _ sin G"B" 
/' ~~ (J' ~ S' ' 

Hence it is evident, that, as soon as the positions of the points 0, C, G" are known, 

the quantities r, /, r", q, ()', q" can be determined. We shall now show how the 

former may be derived from the quantities 

i=P,2(»-+=:-l)r"=«, 

from which, as we have before said, ovu: method started. 



Sect. 1.] three complete observations. 191 

140. 

We first remark, that if N were any point whatever of the great circle CC C"^ 
and the distances of the points C, C, G" from the point N were counted in the 
direction from C to C", so that in general 

NC" — NC'=2f, NG" — NG=^2f\ NG' — ]SfG=2f", 
we shall have 
I. = sin 2/sin NG— sin 2/ sin NG' + sin 2f" sin NG". 

We will now suppose N to be taken in the intersection of the great circles 
BB^B", CC" C", as in the ascending node of the former on the latter. Let us 
denote by (S, (£', S", 2), 2)', 2)", respectively, the distances of the points G, G', G", 
D, D', D" from the great circle BB^B", taken positively on one side, and nega- 
tively on the other. Then sin d, sin %!, sin S", will evidently be proportional to 
siniVC, siniVC, siniVC"', whence equation I. is expressed in the following form: — 

= sin 2/ sin S — sin 2/ sin £' + sin If" sin ^' ; 
or multiplying by rr/', 

n. — nr sin ^ — nr sin g' -f ri'r" sin S". 

It is evident, moreover, that sin £ is to sin '3)', as the sine of the distance of the 
point G from B is to that of D' from B, both distances being measured in the 
same direction. We have, therefore, 

. a' sin %' sin GS 
sm (^AD — Gy 

in precisely the same way, are obtained, 

. ^ sin ID" sin OS 



— sin ^ : 

— sinS": 



sin X sin C'B* sm.lj' sm G' B* 

'' (sTn^'Z)— 3' + (t) sin(^'i7'— S' + ff)' 

sin X sin G"B" sin %' sin G"B' 



sin {A'D — 5") ~ sin {A'U — 6") * 
Dividing, therefore, equation IT. by r" sin (E", there results, 

~ ^^ ' /'sin G"B" ' sin (AD' —d) ^ ' 7^'smG"B" ' sin (A'B—S'-^-a) 



192 DETERMINATION OF AN ORBIT FROM [BoOK II. 

If now we designate the arc G'B' by z, substitute for r, /, /' their values in 
the preceding article, and, for the sake of brevity, put 



[11] 



JR' sm.^' €ra.{Aiy ■ 



L-^^-J R' sin 5" sin {AD _ 3' + a) ~~ *' 
our equation wiU become 
m. = an — ^^/?^^4:=l^+n" 

smz ' 

The coefl&cient I may be computed by the following formula, which is easil;^ 

derived from the equations just introduced : — 

ig^sina'sin(^i7^ — 5) _ , 
L15J a X ^ gjjj 3 gij^ {A!D'—b' + (t) — ^ • 

For verifying the computation, it wiU be expedient to use both the formulas 12 
and 13. When sin(J['Z>" — ^' -[-<?) is greater than sin(J.'i> — d' -^-q), the latter 
formula is less affected by the unavoidable errors of the tables than the former, 
and so wiU be preferred to it, if some small discrepancy to be explained in this 
way should result in the values of i ; on the other hand, the former formula is 
most to be relied upon, when sin [A^D" — d' -\- o) is less than sin [AD — d' -\- o); 
a suitable mean between both values will be adopted, if preferred. The follow- 
ing formulas can be made to answer for examining the calculation; their not very 
difl&cult derivation we suppress for the sake of brevity. 

p. asin(Z" — V) hsm{l" — I) sin (5' — a) |^ sin (Z' — I) 

B R' • si^'fi^ I ~E' ' 

, R' sin 5' fTcos R cos S' 



~~ R" sin 8"* sin {A U — 8) sin «" 

in which (article 138, equation 10,) U expresses the quotient 

S _ r sin (<-!-/) 

sin ((5' — a) cos {d' — a) ' 

141. 

From P = — , and equation HI. of the preceding article, we have 

, , „.P-\-a I ,sin(2: — a) 



Sect. 1.] theee complete observations. 193 

thence, and from 

_ 2 (^' - ly and /= ^^ 

\ «' / sinz 

is obtained, 

Putting, therefore, for the sake of brevity, 

L-^ -1 2E'^sm^8'smG~^' 

and introducing the auxiliary angle w such that 
tan CD = 



3^^ — cosa, 
we have the equation 
IV. e Qsiao) sin* s = sin (^ — w — or), 

from which we must get the unknown quantity s. That the angle w may be 
computed more conveniently, it will be expedient to present the preceding for 
mula for tan w thus : — 

P(- ^) + (- «) 

Vcosff / ' Vcosff / 
"Whence, putting. 



d, 



we shall have for the determination of to the very simple formula, 

tana>-^^^+^ 

We consider as the fourth step the computation of the quantities a, b, c, d, e, 

25 



[16] 
[16] 


a 

cos a 


' 1 

tanff 


* 1 



194 DETERiirXATION OF AN OEBIT FROM [BoOK 11. 

by means of the formulas 11-16, depending on given quantities alone. The 
quantities h, c, e, will not themselves be required, only their logarithms. 

There is a special case in which these precepts require some change. That 
is, when the great circle BB" coincides with A'B", and thus the points B, B'^ 
with D', D, respectively, the quantities a, h would acquire infinite values. Put- 
ting, in this case, 

^ sin d sin {A'jy — S'J^a) 

R'smd's\n{A]y' — d) ~^' 

in place of equation HE. we shall have 

f. n' sin (z — d) 

sm z ' 

whence, making 

, jr sin ff 

tan 0) = p I ., r, 

P-{-(l — ncosa)^ 

the same equation IV. is obtained. 

In the same manner, in the special case when g = 0, c becomes infinite, and 
w = 0, on account of which the factor c sin w, in equation IV., seems to be inde- 
terminate ; nevertheless, it is in reality determinate, and its value is 

J P + g 

2R'HmH' {b—l){P^d)' 

as a little attention wiU show. In this case, therefore, sm s becomes 

142. 

Equation IV., which being developed rises to the eighth degree, is solved by 
trial very expeditiously in its unchanged form. But, from the theory of equa- 
tions, it can be easily shown, (which, for the sake of brevity, we shall dispense 
with explaining more fully) that this equation admits of two or four solutions by 
means of real values. In the former case, one value of sin s will be positive ; 
and the other negative value must be rejected, because, by the nature of the 
pro])lem, it is impossible for / to become negative. In the latter case, among the 
values of sin s one will be positive, and the remaining three negative, — when. 



Sect. 1.] three complete observations. 195 

accordingly, it will not be doubtful which must be adopted, — or three positive 
with one negative ; in this case, from among the positive values those, if there are 
any, are to be rejected which give s greater than d', since, by another essential 
condition of the problem, {/ and, therefore, sin {d' — g), must be a positive quantity. 
When the observations are distant from each other by moderate intervals of 
time, the last case will most frequently occur, in which three positive values of 
sin satisfy the equation. Among these solutions, besides that which is true, 
some one will be found making differ but little from d\ either in excess or 
in defect; this is to be accounted for as follows. The analytical treatment of 
our problem is based upon the condition, simply, that the three places of the heav- 
enly body in space must fall in right lines, the positions of which are determined 
by the absolute places of the earth, and the observed places of the body. Now, 
from the very nature of the case, these places must, in fact, fall in those parts of 
the right lines whence the light descends to the earth. But the anatytical equa- 
tions do not recognize this restriction, and every system of places, harmonizing of 
course with the laws of Kepler, is embraced, whether they lie in these right lines 
on this side of the earth, or on that, or, in fine, whether they coincide with the 
earth itself Now, this last case will undoubtedly satisfy our problem, since the 
earth moves in accordance with these laws. Thence it is manifest, that the equa- 
tions must include the solution in which the points C. C, G" coincide with A, A', A" 
(so long as we neglect the very small variations in the elliptical places of the earth 
produced by the perturbations and the parallax). Equation TV., therefore, must 
always admit the solution s = d', if true values answering to the places of the 
earth are adopted for P and Q. So long as values not differing much from these 
are assigned to those quantities (which is always an admissible supposition, when 
the intervals of the times are moderate), among the solutions of equation TV., 
some one will necessarily be found which approaches very nearly to the value 

For the most part, indeed, in that case where equation IT. admits of three 
solutions by means of positive values of sin s, the third of these (besides the true 
one, and that of which we have just spoken) makes the value of 2 greater than 
d', and thus is only analytically possible, but physically impossible ; so that it can- 



196 DETERMINATION OF AN ORBIT FROM [BoOK II. 

not then be doubtful wbicb is to be adopted. But yet it certainly can happen, 
that the equation may admit of two distinct and proper solutions, and thus that 
our problem may be satisfied by two wholly different orbits. But in such an 
event, the true orbit is easily distinguished from the false as soon as it is possible 
to bring to the test other and more remote observations. 

143. 

As soon as the angle z is got, r is immediately had by means of the equation 

r = ; . 

sinz 

Further, from the equations P = — and III. we obtain, 

n'r' {P-\-d)R'%m8' 

n b sin (z — a) ' 

nV 1 nV 

n" P ' n 

Now, in order that we may treat the formulas, according to which the posi- 
tions of the points C, C", are determined from the position of the point C , in such 
a manner that their general truth in those cases not shown in figure 4 may 
immediately be apparent, we remark, that the sine of the distance of the point 
C from the great circle CB (taken positively in the superior hemisphere, nega- 
tively in the inferior) is equal to the product of sin ^' into the sine of the distance 
of the point C from Z>", measured in the positive direction, and therefore to 

— sin ^" sin CD" = — sin e" sin (s -j- A'D" — d') ; 
in the same manner, the sine of the distance of the point 0" from the same great 
circle is — sin e' sin C"JD\ But, evidently, those sines are as sin CO' to sin CO", or 
as ^, to ^, , or as ti'V to nV. Putting, therefore, CD' = ^", we have 

r smi, =z—^.- — ^sm(s-\- AD — o). 
n" sine ^ ' ^ 

Precisely in the same way, putting CD' = t, is obtained 

TT-T • ;- ^''^ sine • / i a't, s>/\ 

VI. ;• sni (, = — . ^-7 sni (s -4- AD — o ) . 

n sme ^ ' ' 

Vn. ■ r sin (C + AD" — AD') = r" P |^ sin (^ -f Al'D — A'D'). 



Sect. 1.] three complete observations. 197 

By combining equations Y. and VI. with the following taken from article 139, 
Vin. r" sin (C'' — ^'D' + §") = R" sin d", 

IX. r&m{(; — AD'^§) = R^md, 

the quantities t, ^", r, r'jwill be thence derived by the method of article 78. 
That this calculation may be more conveniently effected, it wiU not be unaccept- 
able to produce here the formulas themselves. Let us put 

i?sin d 



[17] 
[18] 



sin {AD' — ( 
E'^md" 



&m{^A'D' — d") 



The computation of these, or rather of their logarithms, yet independent of P 
and Q, is to be regarded as the ffth and last step in the, as it were, preliminary 
operations, and is conveniently performed at the same time with the computation 
of a, b, themselves, or with the fourth step, where a becomes equal to ~ . 
Making, then, 



n sine 

n'r' sin f' 
«" * sin £ 



^, sin [z + A'D — d') =p, 



^{lp—l) = q, 

we derive C and r from r sin ^ = j??, r cos ^ = ^ ; also, ^" and r" from r" sin <i" =^p", 
and r" cos i^" = q". No ambiguity can occur in determining t and 'C'\ because r 
and /' must, necessarily, be positive quantities. The complete computation can, 
if desired, be verified by equation YII. 

There are two cases, nevertheless, where another course must be pursued. 
That is, when the point D' coincides with B, or is opposite to it in the sphere, 
or when AD' — d = or 180°, equations VI. and IX. must necessarily be iden- 



198 DETERailNATION OF AN ORBIT FROM [BoOK 11. 

tical, and we should have k = oo , Ip — 1 = 0, and q, therefore, indeterminate. 
In this case, I," and ;•" will be determined, in the manner we have shown, but 
then C and r must be obtained b j the combination of equation YII. with VI. or 
IX. We dispense with transcribing here the formulas themselves, to be found 
in article 78; we observe, merely, that in the case where AD' — d" is in fact 
neither =: nor = 180°, but is, nevertheless, a very small arc, it is preferable 
to follow the same method, since the former method does not then admit of the 
requisite precision. And, in fact, the combination of equation VII. with VI. or IX. 
will be chosen according as sin [AD" — AD') is greater or less than sin [AD' — d). 

In the same manner, in the case in which the point D', or the one opposite to 
it, either coincides with B" or is little removed from it, the determination of C" 
and 7'' by the preceding method would be either impossible or unsafe. In this 
case, accordingly, t and r will be determined by that method, but l" and r" by 
the combination of equation VII. either with V. or with VIII., according as sin 
[A'D — A'D') is greater or less than sin {A'D^ — d"). 

There is no reason to fear that D' wiU coincide at the same time with the points 
B, B", or with the opposite points, or be very near them ; for the case in which 
B coincides with B", or is but little remote from it, we excluded above, in article 
138, from our discussion. 

144. 

The arcs t, and L." being found, the positions of the points C, C", will be given, 
and it will be possible to determine the distance CC"-= 2/' from C, l" and t. 
Let u, u", be the inclinations of the great circles AB, A"B" to the great circle CO" 
(which in figure 4 will be the angles COD' and 180° — C6'"Z>', respectively), 
and we shall have the following equations, entirely analogous to the equations 
3-6, article 137 : — 

sin/' sin ^ (?f" -j- tt) ^ sin I e' sin ^ (t -|- ^"), 
sin/' cos ^ {u" -j- m) = cos ^ e' sin ^c — (^"), 
cos/' sin i (u" — u) =z sin ^ e' cos 2 (C -[- C")> 
cos/' cos i (u" — u) = cos I e' cos i (; — C"). 



Sect. 1.] three complete observations. 199 

The two former will give ^ {iif'-\-i(,) and sin/', the two latter ^ (^«" — u) and cos/'; 
from sin/' and cos/' we shall have/'. It will be proper to neglect in the first 
hypotheses the angles ^ (?i!"-|-^0 and i [u" — u), which will be used in the last 
hypothesis only for determining the position of the plane of the orbit. 

In the same way, exactly,/ can be derived from £, CD and C"D; also/" 
from e", CD" and O'D" ; but the following formulas are used much more con- 
veniently for this purpose : — 

sin 2 / = r sin 2 /' . -7-7, 
sin 2 /" = r" sin 2/'.'^, 

. . n ri' 

in which the logarithms of the quantities -7-;, -7—, are already given by the pre- 
ceding calculations. Finally, the whole calculation finds a new verification in 
this, that we must have 

2/-|-2/" = 2/'; 
if by chance any difference shows itself, it will not certainly be of any impor- 
tance, if all the processes have been performed as accurately as possible. Never- 
theless, occasionally, the calculation being conducted throughout with seven 
places of decimals, it may amount to some tenths of a second, which, if it appear 
worth while, we may with the utmost facility so distribute between 2 /and 2/" 
that the logarithms of the sines may be equally either increased or diminished, 
by which means the equation 

p r sin 2/' w" 

/'sin2/~^ 

will be satisfied with all the precision that the tables admit. When /and/" differ a 
little, it will be sufficient to distribute that difference equally between 2/ and 2/". 

145. 

After the positions of the heavenly body in the orbit have been determined in 
this manner, the double calculation of the elements will be commenced, both by 
the combination of the second place with the third, and the combination of the 
first with the second, together with the corresponding intervals of the times. 



200 DETERMINATION OF AN ORBIT FROM [BoOK II. 

Before this is undertaken, of course, the intervals of the times themselves require 
some correction, if it is decided to take account of the aberration agreeably to the 
third method of article 118. In this case, evidently, for the true times are to be 
substituted fictitious ones anterior to the former, respectively, by 493(^), 493^', 
493^)" seconds. For computing the distances q, q', i>", we have the formulas : — 

_ Esm(AI>' — t) _ rsm(Aiy — 
^'~sm{^—AD'-\-6)~ smd ' 

, Ii'sm(d'—s) r' sin (S'—z) 

" sin z sin ^ ' 

,,_ m' si n (A"iy — C ) _ /' sin (A" IT — C) 
^ ~sm{Z" — A"iy-\-d")~ smd" 

But, if the observations should at the beginning have been freed from 
aberration by the first or second method of article 118, this calculation may be 
omitted ; so that it will not be necessary to deduce the values of the distances (>, 
(^/, q", unless, perhaps, for the sake of proving that those values, upon which the 
computation of the aberration was based, were sufficiently exact. Finally, it is 
apparent that all this calculation is also to be omitted whenever it is thought 
preferable to neglect the aberration altogether. 

146. 

The calculation of the elements — on the one hand from /, r", 2/ and the 
corrected interval of the time between the second and third observations, the 
product of which multiplied by the quantity k, (article 1,) we denote by &, and 
on the other hand from r, /, If" and the interval of time between the first and 
second observations, the product of which by Ti will be equal to ^" — is to be car- 
ried, agreeably to the method explained in articles 88-105, only as far as the 
quantity there denoted by y, the value of which in the first of these combinations 
we shall call iq, in the latter r['. Let then 

Orf ' r 7^' rirf' cos f cos f cos f" ' 

and it is evident, that if the values of the quantities P, Q, upon which the whole 
calculation hitherto is based, were true, we should have in the result P' = P, 



Sect. 1.] three complete observations. 201 

Q' ■= Q. And conversely it is readily perceived, that if in the result P' = P, 
Q' = Q, the double calculation of the elements from both combinations Avould, if 
completed, furnish numbers entirely equal, by which, therefore, all three observa- 
tions will be exactly represented, and thus the problem wholly satisfied. But 
when the result is not P' = P, Q'= Q, let P'—P, Q' — Q be taken for X and Y, 
if, indeed, P and Q were taken for x and y; it will be still more convenient to put 

logP = ^,log ^ = ^,logP'— logP = X,log ^'_log ^= Z , 
Then the calculation must be repeated with other values of x, y. 

147. 

Properly, indeed, here also, as in the ten methods before given, it would be 
arbitrary what new values we assume for x and y in the second hypothesis, if 
only they are not inconsistent with the general conditions developed above ; but 
yet, since it manifestly is to be considered a great advantage to be able to set out 
from more accurate values, in this method we should act with but little prudence 
if we were to adopt the second values rashl}^, as it were, since it may easily be 
perceived, from the very nature of the subject, that if the first values of P and Q 
were affected with slight errors, P' and Q' themselves would represent much more 
exact values, supposing the heliocentric motion to be moderate. Wherefore, we 
shall always adopt P' and Q' themselves for the second values of P and Q, or 
log P', log Q' for the second values of x and y, if log P, log Q are supposed to 
denote the first values. 

Now, in this second hypothesis, where all the preliminary work exhibited 
in the formulas 1-20 is to be retained without alteration, the calculation will be 
undertaken anew in precisely the same manner. That is, first, the angle o> 
will be determined; after that z, r, ^, !!^, c, r, i;", r", f, f, f". From the dif- 
ference, more or less considerable, between the new values of these quantities 
and the first, a judgment will easily be formed whether or not it is worth while 
to compute anew the correction of the times on account of aberration ; in the 
latter case, the intervals of the times, and therefore the quantities & and &", wiU 
remain the same as before. Finally, i], if are derived from /, r, r"J", r, r and 

26 



202 DETERMINATION OF AN ORBIT FROM [BoOK 11. 

the intervals of the times ; and hence new values of P' and Q', which commonly 
differ much less from those furnished by the first hypothesis, than the latter from 
the original values themselves of P and Q. The second values of X and Y will, 
therefore, be much smaller than the first, and the second values of P', Q', will be 
adopted as the third values of P, Q, and with these the computation will be 
resumed anew. In this manner, then, as from the second hypothesis more exact 
numbers had resulted than from the first, so from the third more exact numbers 
will again result than from the second, and the third values of P', Q' can be taken 
as the fourth of P, Q, and thus the calculation be repeated until an hypothesis 
is arrived at in which X. and Y may be regarded as vanishing ; but when the 
third hypothesis appears to be insufficient, it will be preferable to deduce the val- 
ues of P, Q, assumed in the fourth hypothesis from the first three, in accordance 
with the method explained in articles 120, 121, by which means a more rapid 
approximation will be obtained, and it will rarely be requisite to go forward to 
the fifth hypothesis. 

148. 

When the elements to be derived from the three observations are as yet 
wholly unknown (to which case our method is especially adapted), in the first 
hypothesis, as we have already observed, — , ^ d", are to be taken for approximate 
values of P and Q, where ^ and ^" are derived for the present from the intervals 
of the times not corrected. If the ratio of these to the corrected intervals is 
expressed by /^ : 1 and u" : 1, respectively, we shall have in the first hypothesis, 

X = log fii — log //' -|- log tj — log rj'^ 

Y= log u -\- log jLi" — log 1] — log if -\- Comp. log cos/ -\- Comp. log cos /' 
-J- Comp. log cos/" -|- 2 log r — log r — log r". 

The logarithms of the quantities ,a, fjf', are of no importance in respect to the re- 
maining terms ; log 7^ and log ?j", which are both positive, in X cancel each other 
in some measure, whence X possesses a small value, sometimes positive, some- 
times negative ; on the other hand, in Y some compensation of the positive terms 
Comp. log cos/, Comp. log cos/', Comp. log cos/" arises also from the negative 



Sect. 1.] three complete observations. 203 

terms log i], log rf', but less complete, for the former greatly exceed the latter. In 
general, it is not possible to determine any thing concerning the sign of log -^. 

Now, as often as the heliocentric motion between the observations is small, it 
will rarely be necessary to proceed to the fourth hypothesis ; most frequently the 
third, often the second, will afford sufficient precision, and we may sometimes be 
satisfied with the numbers resulting from even the first hypothesis. It will be 
advantageous always to have a regard to the greater or less degree of precision 
belonging to the observations; it would be an ungrateful task to aim at a pre- 
cision in the calculation a hundred or a thousand times greater than that which 
the observations themselves allow. In these matters, however, the judgment is 
sharpened more by frequent practical exercise than by rules, and the skilful 
readily acquire a certain faculty of deciding where it is expedient to stop. 

149. 

Lastly, the elements themselves will be computed in the final hypothesis, 
either from/, r', r", or from/", r, /, carrying one or the other of the calculations 
through to the end, which in the previous hypotheses it had only been requisite 
to continue as far as rj^rj"; if it should be thought proper to finish both, the 
agreement of the resulting numbers will furnish a new verification of the whole 
work. It is best, nevertheless, as soon as /,/',/", are got, to obtain the elements 
from the single combination of the first place with the third, that is, from f^,r, r", 
and the interval of the time, and finally, for the better confirmation of the com- 
putation, to determine the middle place in the orbit by means of the elements 
found. 

In this way, therefore, the dimensions of the conic section are made known, 
that isy the eccentricity, the semi-axis major or the semi-parameter, the place 
of the perihelion with respect to the heliocentric places C, C\ G" , the mean 
motion, and the mean anomaly for the arbitrary epoch if the orbit is elliptical, or 
the time of perihelion passage if the orbit is hyperbolic or parabolic. It only 
remains, therefore, to determine the positions of the heliocentric places in the 
orbit with respect to the ascending node, the position of this node with reference 
to the equinoctial point, and the inclination of the orbit to the ecliptic (or the 



204 DETERMINATION OF AN ORBIT FROil [BoOK II. 

equator). AH this may be efifected by the solution of a single spherical tri- 
angle. Let Q be the longitude of the ascending node ; i the inclination of the 
orbit ; (/ and ff" the arguments of the latitude in the first and third observations ; 
lastly, let I — 9,= h, I" — S2 = W. Calling, in figure 4, 9, the ascending node, 
the sides of the triangle S^ilCwill be AD' — t, ^, li, and the angles opposite to 
them, respectively, i, 180° — y,ii. We shall have, then, 

sin i 2 sin i (^ -|- li) = sin ^ {AJ}' — Q sin 2 (/ -|- u) 
sin h i cos 2 [g -\- h) = cos ^ [AD' — 'Q) sin ^ (7 — u) 
cos h i sin ^ {g — h) = sin i (AI)' — C) cos 2 (/ -j- «) 
cos J ^ cos ^ {g — h) = cos ^ [AD' — ;) cos i (/ — ^«). 
The two first equations will give ^ (^-f-A) and sin ^ i, the remaining two ^ {g — h) 
and cos ^ 2 ; from y will be known the jjlace of the perihelion with regard to the 
ascending node, from h the place of the node in the ecliptic ; finally, ^ will be- 
come known, the sine and the cosine mutually verifying each other. We can 
arrive at the same object by the help of the triangle 9,A"C', in which it is only 
necessary to change in the preceding formulas the symbols g, h, A, L, /, tt into g", 
h", A", 'C", y", li' . That still another verification may be provided for the whole 
work, it mil not be unserviceable to perform the calculation in both ways; 
when, if any very slight discrepancies should show themselves between the values 
of 2^ 9, and the longitude of the perihelion in the orbit, it will be proper to take 
mean values. These difierences rarely amount to OM or 0^2, provided all the 
computations have been carefully made with seven places of decimals. 

When the equator is taken as the fmidamental plane instead of the ecliptic, 
it will make no difference in the computation, except that in place of the points 
A, A" the intersections of the equator with the great cu'cles AB, A.'B" are to be 
adopted. 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



205 



150. 

We proceed now to the illustration of this method by some examples fully 
explained, which will show, in the plainest manner, how generally it applies, and 
how conveniently and expeditiously it leads to the desired result.* 

The new planet Juno will furnish us the first example, for which purpose we 
select the following observations made at Greenwich and communicated to us by 
the distinguished Maskelyne. 



Mean Time, Greenwich. 


App. Eight Ascension. 


App. Declination S. 


1804, Oct. 5 10^ 51™ 6' 
17 9 58 10 
27 9 16 41 


357° 10' 22".35 
355 43 45 .30 
355 11 10 .95 


6° 40' 8" 

8 47 25 

10 2 28 



From the solar tables for the same times is found 





Longitude of the Sun 
from App. Equin. 


Nutation. 


Distance from 
the Earth. 


Latitude of 
the Sun. 


Appar. Obhquity of 
the Ecliptic. 


Oct. 5 
17 

27 


192° 28' 53".72 
204 20 21 .54 
214 16 52 .21 


4- 15".43 
4-15 .51 
-|-15 .60 


0.9988839 
0.9953968 
0.9928340 ■ 


— 0".49 
+ 0.79 

— 0.15 


23° 27' 59".48 
59.26 
59 .06 



We will conduct the calculation as if the orbit were wholly unknown : for 
which reason, it will not be permitted to free the places of Juno from parallax, 
but it will be necessary to transfer the latter to the places of the earth. Accord- 
ingly we first reduce the observed places from the equator to the ecliptic, the 
apparent obliquity being employed, whence results, 



* It is incorrect to call one method more or less exact than another. That method alone can be con- 
sidered to have solved the problem, by which any degree of precision whatever is, at least, attainable. 
"Wherefore, one method excels another in this respect only, that the same degree of precision may be 
reached by one more quickly, and with less labor, than by the other. 



206 



DETERMINATION OF AN ORBIT FROM 



[Book II. 





App. Longitude of Juno. 


App. Latitude of Juno. 


Oct. 5 

17 
27 


354° 44' 54".27 
352 34 44.51 
351 34 51 .57 


— 4°59'31".59 

— 6 21 56.25 

— 7 17 52.70 



We join directly to this calculation the determination of the longitude and 
latitude of the zenith of the place of observation in the three observations : the 
right ascension, in fact, agrees with the right ascension of Juno (because the 
observations have been made in the meridian) but the declination is equal to the 
altitude of the pole, 51° 28' 39". Thus we get 





Long, of the Zenith. 


Lat. of the Zenith. 


Oct. 5 
17 

27 


24° 29' 
23 25 
23 1 


46° 53' 
47 24 
47 36 



Now the fictitious places of the earth in the plane of the ecliptic, from which 
the heavenly body would appear in the same manner as from the true places of 
the observations, will be determined according to the precepts given in article 72. 
In this way, putting the mean parallax of the sun equal to 8".6, there results, 





Reduction of Longitude. 


Beduction of Distance. 


Reduction of Time. 


Oct. 5 
17 
27 


— 22" .39 

— 27 .21 

— 35 .82 


-1-0.0003856 
4- 0.0002329 
4- 0.0002085 


— 0M9 

— .12 

— .12 



The reduction of the time is added, only that it may be seen that it is wholly 
insensible. 

After this, all the longitudes, both of the planet and of the earth, are to be 
reduced to the mean vernal equinox for some epoch, for which we shall adopt 
the beginning of the year 1805 ; the nutation being subtracted the precession is 
to be added, which, for the three observations, is respectively 11".87, 10".23, 8".86, 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



207 



so that — 3".56 is to be added for the first observation, — 5".28 for the second, 
— 6^74 for the third. 

Lastly the longitudes and latitudes of Juno are to be freed from the aberra- 
tion of the fixed stars ; thus it is found by well-known rules, that we must sub- 
tract from the longitudes respectively 19''.12, 17".ll, 14".82, but add to the lati- 
tudes 0".53, r'.18, 1".75, by which addition the absolute values are diminished, 
since south latitudes are considered as negative. 



151. 

All these reductions being properly applied, we have the correct data of the 
problem as follows : — 
Times of the observations reduced 

to the meridian of Paris . Oct. 5.458644 17.421885 27.393077 

Longitudes of Juno, a, a', a" . 354°44'3r.60 352°34'22''.12 35r34'30".01 

Latitudes, (^, (f, §" _ 4 59 31 .06 —6 21 55 .07 —7 17 50 .95 

Longitudes of the earth, I, I', I" 12 28 27 .76 24 19 49 .05 34 16 9 .65 

Logs, of the distances, R, R', R" 9.9996826 9.9980979 9.9969678 

Hence the calculations of articles 136, 137, produce the following numbers, 



7,7,7 

d,d',d" 

logarithms of the sines 
AD, AD', AD" . . 
A'D,A:'D',AD" . . 



196"= 


0' 8' 


.36 


18 


23 59 


.20 




9.4991995 


232 


6 26 


.44 


241 


51 15 


.22 


2 


19 34 


.00 




8.6083885 



logarithms of the sines . . 

log sin ^ e' ...... 

log cos ^ g' 

Moreover, according to article 138, we have 
log tan/? .... 8.9412494 ra log tan ^" . . 
log sin (a"—/') . 9.7332391 w logsin(a — /') 
log cos («" — /') . 9.9247904 log cos (a— r) 



191° 58' 0".33 

32 19 24 .93 

9.7281105 

213 12 29 .82 

234 27 .90 

7 13 37 .70 

9.0996915 

8.7995259 

9.9991357 



190°4r40M7 

43 11 42 .05 

9.8353631 

209 43 7 .47 

221 13 57 .87 

4 55 46 .19 

8.9341440 



9.1074080 J^ 
9.6935181^2 
9.9393180 



208 



DETERMINATIOX OF AN ORBIT FROM 



[Book II. 



Hence 

log (tan /5 cos {a" — V) — tan ^" cos {a — /')) = log Tsm t 8.6786513 
logsm(«" — a)=logrcosi^ 8.7423191 « 

Hence t = 145° 32' 57".78 
!{_|-/ = 337 30 58.11 



log ^ . . . 

log sin it -|- y') 



8.8260683 
9.5825441 »^ 



Lastly 
log (tan /? sin {a!' — I) — tan /3" sin (a — /)) 
\o^Tmv{t^Y) 



\o^8 



8.2033319 ?? 
8.4086124 w 



9.4075427 w " 
9.5050667 ?2 
9.5376909 w 
9.2928554 w 
9.2082723^2 

a=-f 0.3543592 



log cos 9.9915661 7z 
« « 9.9853301 n 



whence log tan (d' — (T) 9.7947195 

8'—a = 31° 56' ir.81, and therefore (7 = 0° 23' 13".12. 
According to article 140 we have 
riy — d" = 191° 15'18".85 log sin 9.2904352^2 
AD'—d =194 48 30 .62 

A"D—r =198 39 33 .17 
A'D —d'-}-a = 200 10 14 .63 
AD'' — d =191 19 8 .27 

A[D"—d' + (T = 189 17 46 .06 
Hence follow, 

loga . . . 9.5494437; 
log J . . . 9.8613533. 
Formula 13 would give log h = 9.8613531, but we have preferred the former 
value, because sin {AD — d' -)- o) is greater than sin {AD" — d' -{-o). 
Again, by article 141 we have, 

3 log i?' sin ^' . . . 9.1786252 

log 2 0.3010300 

log sin a 7.8295601 

2.6907847 



7.3092163 and therefore log c 



log 5 9.8613533 

loffcoscr 9.9999901 



9.8613632 



Sect. 1.] theee complete observations. 209 

whence = 0.7267135. Hence are derived 

d = — 1.3625052, log e = 8.3929518 n 
Finally, by means of formulas, article 148, are obtained, 
logjt . . . . 0.0913394 ?2 
log;t" .... 0.5418957 w 
logil . . . . 0.4864480 w 
logr .... 0.1592352 w 

152. 

The preUminary calculations being despatched in this way, we pass to the 
first hypothesis. The interval of time (not corrected) between the second and 
third observations is 9.971192 days, between the first and second is 11.963241. 
The logarithms of these numbers are 0.9987471, and 1.0778489, whence 

log ^ = 9.2343285, log ^" = 9.3134303. 
"We will put, therefore, for the first ht/pothedsy 
:?;=: log P= 0.0791018 
^== log ^=8.5477588 
Hence we have P = 1.1997804, P + a = 1.5541396, P-\-d= — 0.1627248 ; 
logg . . . 8.3929518 w 
log(P + a). 0.1914900 
C.log{F^d) 0.7885463 w 

log tan 0) . . 9.3729881, whence w = + 13°16'5r.89, w + = -f 13°40' 5".01. 
log^ . . . 8.5477588 
logc . . . 2.6907847 
log sin CO . . 9.3612147 
log ^c sin w . 0.5997582 

The equation 

^c sin w sin^ ^= sin (^ — 13°40' 5".01) 
is found after a few trials to be satisfied by the value s=: 14°35' 4".90, whence 
we have log sin s = 9.4010744, log / = 0.3251340. That equation admits of three 
other solutions besides this, namely, 

27 



210 DETERISimATION OF AN ORBIT FROM [BoOK 11. 

s = 32° 2' 28" 

0=137 27 59 

= 193 4 18 
The third must be rejected because sui z is negative ; the second because z is 
greater than d' ; the first answers to an approximation to the orbit of the earth 
of which we have spoken in article 142. 

Further, we have, according to article 143, 

log^^ 9.8648551 

log (P -fa) 0.1914900 

C. log sin (s — a). . . . 0.6103678 

log'^ 0.6667029 

logP 0.0791018 

log^ 0.5876011 

^-^A'D — d' = z ^199° ^r r.51 =: 214" 22' 6".41; log sin = 9.7516736 ?z 
z^A'D"—d'=z-{-lSS 54 32 .94 = 203 29 37 .84; log sin = 9.6005923 w 
Hence we have log^= 9.9270735 n, log /'= 0.0226469^, and then 

log q = 0.2930977 n, log q" = 0.2580086 n, 
whence result 

C = 203° 17' 3r.22 log r = 0.3300178 
r=110 10 58 .88 logr"= 0.3212819 
Lastly, by means of article 144, we obtain 

^u"-\-u)= 205° 18' 10".53 
^(i/'_m)= — 3 14 2 .02 
/'= 3 48 14 .66 
log sin 2/ . . . 9.1218791 log sin 2/' . . . 9.1218791 

logr 0.3300178 log/' 0.3212819 

Clog— .... 9.3332971 C.log^ .... 9.4123989 

log sin 2/ . . . 8.7851940 lo^ si^ 2/" . . . 8^8^555599 
2/= 3°29"46'.03 2/"= 4°6'43".28 

The sum 2/ + 2/" differs in this case from 2/ only by 0".01. 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



211 



Now, in order tliat the times may be corrected for aberration, it is necessary to 
compute the distances q, q', q" by the formulas of article 145, and afterwards to 
multiply them by the time 493*, or 0*^.005706. The following is the calculation, 

logr" .... 0.32128 
logsin(^''i>'—r) 9.61384 
Clog sin r . . 0.16464 



logr . . . . 0.33002 
logsin(^Z>'— C) 9.23606 
C.loffsin^ . . 0.50080 



log/ 






0.32513 


log sin [d' — 


■^) 


9.48384 


Clog 


sind"' 




0.27189 



log^ • 
log; const. 



0.06688 
7.75633 



log) 



0.08086 
7.75633 



q" . . . . 0.09976 
7.75633 



log of reduction 7.82321 


7.83719 


7.85609 


reduction = 


0.006656 


0.006874 


0.007179 


Observations. 


rnn-ected times. 


Intervals. 


Logarithms. 


I. 

n. 
in. 


Oct. 5.451988 
17.415011 

27.385898 


11^.963023 

9.970887 


1.0778409 
0.9987339 



The corrected logarithms of the quantities ^, ^", are consequently 9.2343153 and 
9.3134223. By commencing now the determination of the elements from /, /, 
r", & we obtain log tj =^ 0.0002285, and in the same manner from /", r, /, ^"we 
get log tj'' = 0.0003191. "We need not add here this calculation explained at 
length in section III. of the first book. 
Finally we have, by article 146, 

logr .... 9.3134223 

C.log^ . . . . 0.7656847 

log 7? .... 0.0002285 

C.logV' . . . 9.9996809 

logP' .... 0.0790164 



2 log/ . . . 


0.6502680 


C. log rr" . . 


9.3487003 


log^r . . . 


8.5477376 


QAogriri" . . 


9.9994524 


Clog cos/ . . 


0.0002022 


C log cos/' . . 


0.0009579 


Clog cos/' . 


0.0002797 


log^'. . . . 


8.5475981 



The first hypothesis, therefore, results in X = — 0.0000854, Y. 



0.0001607. 



212 



DETEEJtllNATION OF Ali ORBIT FROM 



[Book II. 



153. 

In the second hypothesis we shall assign to P, Q, the very values, which in the 
first we have found for P', §'. We shall put, therefore, 

a? == log P = 0.0790164 
y = log ^ = 8.5475981 

Since the calculation is to be conducted in precisely the same manner as in 
the first hypothesis, it will be sufficient to set down here its principal results : — 



to . , . 

log § c sin w 

s . . . 

loo- / . . 



1 n r 



log 



n' r' 



13°15'38'a3 

13 38 51 .25 

0.5989389 

14 33 19 .00 
0.3259918 
0.6675193 

0.5885029 
203 16 38 .16 



r . . 

log r . . 
log r" 

I {u" -{- u) 
h{u" — u) 
2/ . . 
2/ . . 

2r . . 



210° 8'24".98 

0.3307676 

0.3222280 

205 22 15 .58 

— 3 14 4 .79 

7 34 53 .32 

3 29 .18 

4 5 53 .12 



It would hardly be worth while to compute anew the reductions of the times 
on account of aberration, for they scarcely differ V from those which we have 
got in the first hypothesis. 

The further calculations furnish log7jz=z 0.0002270, log^" = 0.0003173, whence 
are derived 

log P'= 0.0790167 X= + 0.0000003 

log ^'=8.5476110 F= + 0.0000129 

From this it appears how much more exact the second hypothesis is than the 
first. 



154. 

In order to leave nothing to be desired, we will still coUvStruct the third hypothe- 
s, in which we shall again choose the values of P', Q', obtained in the second 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



213 



hypothesis, as the values of P, Q. Putting, therefore, 

:?; = log P = 0.0790167 

y = log Q = 8.5476110 
the following are found to be the principal results of the calculation : — 



oj -\- a . 
log Qosino) 



log / 



log 



n r 
n'r' 



13°15'38'^39 

13 38 51 .51 
0.5989542 

14 33 19 .50 
0.3259878 
0.6675154 

0.5884987 
203 16 38 .41 



r . . 

logr . . 
log r" 
h {u" -[- u) 
h {u" — u) 
2/ . . 
2/ . . 
2/" . . 



210° 8'25''.65 

0.3307640 

0.3222239 

205 22 14 .57 

— 3 14 4 .78 

7 34 53 .73 

3 29 .39 

4 5 53 .34 



All these numbers differ so little from those which the second hypothesis fur- 
nished, that we may safelj^ conclude that the third hypothesis requires no further 
correction.* We may, therefore, proceed to the determination of the elements 
from 2/', r, /', &\ which we dispense with transcribing here, since it has already 
been given in detail in the example of article 97. Nothing, therefore, remains 
but to compute the position of the plane of the orbit by the method of article 
149, and to transfer the epoch to the beginning of the year 1805. This computa- 
tion is to be based upon the following numbers : — 

AD' — i;= 9°55'5r.41 
i(/-f«) = 202 18 13 .855 
i(;/— m)=— 6 18 5 .495 



whence we obtain 



l{g^h)= 196°43'14".62 
j(^_/^)=:_4 37 24 .41 
hi = 6 33 22.05 



* If the calculation should be carried through in the same manner as in the preceding hypotheses, 
we should obtain X=zO, and Z=-[~^'^0^*^'^0^' which value must be regarded as vanishing, and, 
in fact, it hardly exceeds the uncertainty always remaining in the last decimal place. 



214 DETERMINATION OF AN OKBIT FROM [BoOK II. 

We have, therefore, h = 201° 20' 3r.03, and m9>=l — h = 171° 7' 48".73 ; fur- 
ther, g = 192° 5' 50".21, and hence, since the true anomaly for the first oloserva- 
tion is found, in article 97, to be 310°55'29".64, the distance of perihelion from 
the ascending node in the orbit, 241° 10' 20".57, the longitude of the perihelion 
62° 18' 9".30 ; lastly, the inclination of the orbit, 13° 6' 44".10. If we prefer to 
proceed to the same calculation from the third place, we have, 

A:'D' — 'Q" = 24° 18' 35".25 

i (/'+«") =196 24 54 .98 

h{7"—u")= — ^ 43 14 .81 
Thence are derived 

^ (/' J^ h") ^ 211° 24' 32".45 
!(/' — //')= — 11 43 48 .48 
a = 6 33 22 .05 

and hence the longitude of the ascending node, l" — A" = 171° 7'48".72, the lon- 
gitude of the perihelion 52° 18' 9".30, the inclination of the orbit 13° 6'44".10, 
just the same as before. 

The interval of time from the last observation to the beginning of the year 
1805 is 64.614102 days; the mean heliocentric motion corresponding to which is 
53293".66 =14° 48' 13".66 ; hence the epoch of the mean anomaly at the begin- 
ning of the year 1805 for the meridian of Paris is 349° 34' 12".38, and the epoch 
of the mean longitude, 41° 52' 21".68. 

155. 

That it may be more clearly manifest what is the accuracy of the elements 
just found, we will compute from them the middle place. For October 17.415011 
the mean anomaly is found to be 332° 28' 54".77, hence the true is 315° 1' 23".02 
and log r", 0.3259877, (see the examples of articles 13, 14); this true anomaly 
ought to be equal to the true anomaly in the first observation increased by the 
angle 2/", or to the true anomaly in the third observation diminished by the 
angle 2/, that is, equal to 315° 1' 22".98; and the logarithm of the radius vector 
should be 0.3259878 : the differences are of no consequence. K the calculation 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



215 



for the middle observation is continued to the geocentric place, the results dif- 
fer from observation only by a few hundredths of a second, (article 63 ;) these 
differences are absorbed, as it were, in the unavoidable errors arising from the 
want of strict accuracy in the tables. 

"We have worked out the preceding example with the utmost precision, to 
show how easily the most exact solution possible can be obtained by our method. 
In actual practice it will rarely be necessary to adhere scrupulously to this 
type. It will generally be sufficient to use six places of decimals throughout; 
and in our example the second hypothesis would have given results not less accu- 
rate than the third, and even the first would have been entirely satisfactory. We 
imagine that it will not be unacceptable to our readers to have a comparison of 
the elements derived from the third hypothesis with those which would result 
from the use of the second or first hypothesis for the same object. We exhibit 
the three systems of elements in the following table : — 





From hypothesis III. 


From hypothesis II. 


From hypothesis I. 


Epoch of mean long. 1805 
Mean daily motion . . 
Perihelion 

Log of semi-axis major . 
Ascending node . . . 
Inclination of the oi-bit . 


41°52'21".68 

824".7989 

52 18 9 .30 

14 12 1 .87 

0.4224389 

171 7 48 .73 

13 6 44 .10 


41°52'18".40 

824".7983 

52 18 6 .66 

14 11 59 .94 

0.4224392 

171 7 49 .15 

13 6 45 .12 


42°12'37".83 

823".5025 

52 41 9 .81 

14 24 27 .49 

0.4228944 

171 5 48 .86 

13 2 37 .50 



By computing the heliocentric place in orbit for the middle observation from 
the second system of elements, the error of the logarithm of the radius vector is 
found equal to zero, the error of the longitude in orbit, O'^.OS ; and in comput- 
ing the same place by the system derived from the first hypothesis, the error of 
the logarithm of the radius vector is 0.0000002, the error of the longitude in 
orbit, 1".31. And by continuing the calculation to the geocentric place we have, 



216 



DETERMINATION OF AN ORBIT FROil 



[Book 1] 





From hypothesis II. 


From hypothesis I. 


Geocentric longitude 

Error 

Geocentric latitude . 
Error 


352° 34' 22".26 

0.14 

6 21 55 .06 

0.01 


352° 34' 19".97 

2.15 

6 21 54.47 

0.60 



156. 

We shall take the second example from Pallas, the following observations of 
which, made at Milan, we take from von Zach's Monatliche Correspondens, Vol. 
XIV., p. 90. 



Mean Time, Milan. 


App. Right Ascension. 


App. Declination S. 


1805, Nov. 5''14'' U'" 4' 
Dec. 6 11 51 27 

1806, Jan. 15 8 50 36 


78° 20' 37".8 
73 8 48.8 
67 14 11 .1 


27° 16' 56".7 
32 52 44.3 
28 38 8.1 



We will here take the equator as the fundamental plane instead of the 
ecliptic, and we will make the computation as if the orbit were still wholly un- 
known. In the first place we take from the tables of the sun the following data 
for the given dates : — 





Longitude of the Sun 
from mean Equinox. 


Distance from 
the Earth. 


Latitude of 
the Sun. 


Nov. 5 
Dec. 6 
Jan. 15 


223° 14' 7".61 
254 28 42 .59 
295 5 47 .62 


0.9804311 
0.9846753 
0.9838153 


+ 0".59 
+ 0.12 
— 0.19 



We reduce the longitudes of the sun, the precessions -}-7".59, +3".36, — 2".ll, 
being added, to the beginning of the year 1806, and thence we afterwards derive 
the right ascensions and declinations, using the mean obliquity 23° 27' 53".53 and 
taking account of the latitudes. In this way we find 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



217 





Eight ascension of the Sun. 


Decl. of the Sun S. 


Nov. 5 
Dec. 6 
Jan. 15 


220° 46' 44".65 
253 9 23 .26 
297 2 51 .11 


15° 49' 43".94 
22 33 39 .45 
21 8 12 .98 



These places are referred to the centre of the earth, and are, therefore, .to be 
reduced by applying the parallax to the place of observation, since the places of 
the planet cannot be freed from parallax. The right ascensions of the zenith to 
be used in this calculation agree with the right ascensions of the planet (because 
the observations have been made in the meridian), and the declination will be 
throughout the altitude of the pole, 45° 28'. Hence are derived the followiag 
numbers : — 





Eight aso. of the Earth. 


Decl. of the Earth N. 


Log of dist. from the Sun. 


Nov. 5 
Dec. 6 
Jan. 15 


40° 46' 48". 51 

73 9 23 .26 

117 2 46.09 


15° 49' 48".59 
22 33 42 .83 
21 8 17 .29 


9.9958575 
9.9933099 
9.9929259 



The observed places of Pallas are to be freed from nutation and the aberra- 
tion of the fixed stars, and afterwards to be reduced, by applying the precession, 
to the beginning of the year 1806. On these accoijnts it will be necessary to 
apply the following corrections to the observed places : — 




28 



218 



DETERMNATION OF AN ORBIT FROM 



[Book IT. 



Hence we have the following places of Pallas, for the basis of the compu- 
tation : — 



Mean Time, Paris. 


Eight Ascension. 


DecHnation. 


Nov. 5.574074 
36.475035 
76.349444 


78° 20' 12".24 
73 8 16 .16 
67 13 40 .93 


— 27° 17' 9".05 

— 32 52 48.96 

— 28 38 2.42 



157. 

Now in the first place we will determine the positions of the great circles 
drawn from the heliocentric places of the earth to the geocentric places of the 
planet. We take the symbols % St', %", for the intersections of these circles 
with the equator, or, if you please, for their ascending nodes, and we denote the 
distances of the points B, B' , B" from the former points by A, A', A". In the 
greater part of the work 'it will be necessary to substitute the symbols 2t, §(', W, 
for A, A', A", and also A, A', A" for ^, 8', d" ; but the careful reader will readily 
understand when it is necessary to retain A, A', A", d, d', d", even if we fail to 
advise him. 

The calculation being made, we find 
Right ascensions of the 
points % r, W . . . 233° 54' 57".10 

r, /, f 51 17 15 .74 

A, A\ A" 215 58 49 .27 

56 26 34 .19 
2t'Z>, %D', %D" ... 23 54 52 .13 
2l"i>, 5t"i>', %:D" . . 33 3 26 .35 
47 1 64 .69 
9.8643525 



£,!;,£ 

logarithms of the sines 

log sin J e' 

log cos i «' .... 



253° 8'57".01 
90 1 3 .19 

212 52 48 .96 
55 26 31 .79 

30 18 3 .25 

31 59 21 .14 
89 34 57 .17 

9.9999885 
9.8478971 
9.8510614 



276°40'25".87 

131 59 58 .03 

220 9 12 .96 

69 10 57 .84 

29 8 43 .32 

22 20 6 .91 

42 33 41 .17 

9.8301910 



Sect. 1.] three complete observations. 219 

The right ascension of the point W is used in the calculation of article 138 
instead of l\ In this manner are found 

log I^ sin?! ..... 8.4868236 w 
log I' cos ^ 9.2848162 w 

Hence t = 189° 2'48".83, log T= 9.2902527 ; moreover, i-{-/ = 279° 3'52".02, 

log /S' 9.0110566 n 

logTsm{t-\-/) . • . 9.2847950^2 
whence J'—a = 208° V 55^^64, and (7 = 4° 50' 53".32. 

In the formulas of article 140 sin d, sin d', sin d" must be retained instead of 
a, h and -, and also in the formulas of article 142. For these calculations we 
have 

WD' — J" = 171° 50' 8".18 log sin 9.1523306 log cos 9.9955759^2 

%j)'—A =174 19 13.98 « « 8.9954722 « « 9.9978629?? 

WD— A" =172 54 13.39 « « 9.0917972 

<^B —J'-\-o = 175 52 56 .49 « « 8.8561520 

^D" — A = 173 9 54 .05 « « 9.0755844 

ri/'— ^' + a =174 18 11 .27 « « 8.9967978 
Hence we deduce 

log;t =0.9211850, logl = 0.0812057 ?2 

log x"= 0.8112762, log r= 0.031969172 

log a = 0.1099088, az=-\- 1.2879790 

log^ =0.1810404, 
log- =0.0711314, 

whence we have log b = 0.1810402. We shall adopt log 5 = 0.1810403 the 
mean between these two nearly equal values. Lastly we have 

log c = 1.0450295 

d = -\- 0.4489906 
log e = 9.2102894 

with which the preliminary calculations are completed. 



220 



DETERMINATION OF AN ORBIT FROM 



[Book II. 



The interval of time between the second and third observations is 39.874409 
days, between the first and second 30.900961 : hence we have 
log & = 9.8362757, log&"= 9.7255533. 
We put, therefore, for the Jirst hf/jyothesis, 

a; = log P=r 9.8892776 
y=. log ^=9.5618290 
The chief results of the calculation are as follows : — 
aj + <7r=20° 8'46".72 
log ^c sin w = 0.0282028 
Thence the true value of s is 21°ir24''.30, and of log/, 0.3509379. The three 
remaining values of s satisfying equation IV., article 141, are, in this instance, 

2= 63° 41' 12" 
^ = 101 12 58 
g=199 24 7 

the first of which is to be regarded as an approximation to the orbit of the earth, 
the deviation of which, however, is here much greater than in the preceding 
example, on account of the too great interval of time. The following numbers 
result from the subsequent calculation : — 
C . . . 

r . . 

log r . . 
log/' . 

^K + ^0 
h {u" — u) 
2/ . . 
2/ . . 
2/' ■ . 



195° 12' 2".48 

196 57 50 .78 

0.3647022 

0.3355758 

266 47 50 .47 

-43 39 5 .33 

22 32 40 .86 

13 5 41 .17 

9 27 .05 



We shall distribute the difference between 2/' and 2/-|-2/", which in this case 
is 0".36, between 2/ and 2/" in such a manner as to make 2/= 13° 5'40".96, 
and2/"=9°26'59".90. 

The times are now to be corrected for aberration, for which purpose we are to 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



221 



put in the formulas of article 145, 
We have, therefore, 



r=ri>'— //"+. 



logr . . . 


0.36470 


log/ 


. . 


. 0.35094 


log sin {AD' — 


C) 9.76462 


log sin ((5'' — 


-s) 9.75038 


Clog sin d . 


0.07918 


Clog 


sin(^' 


. 0.08431 


log const. . . 


7.75633 
7.96483 


log const. . 


. 7.75633 








7.94196 


reduction of"! 
the time J 


0.009222 






0.008749 


Hence follow. 










Observations. 


Corrected times. 




Intervals. 


I. 

n. 
in. 


Nov. 5.564852 
36.466286 
76.340252 




30''.901434 

39.873966 



log/' .... 0.33557 
logsin(yl"X>'—r) 9.84220 
Clog sin (^" . . 0.02932 
log const. . . 7.75633 



7.96342 
0.009192 



Logarithms. 

1.4899785 
1.6006894 



whence are derived the corrected logarithms of the quantities &, 6" respectively 
9.8362708 and 9.7265599. Beginning, then, the calculation of the elements 
from /, /', 2/, 6, we get log rj = 0.0031921, just as from r, r, 2f", ^" we obtain 
log ri" = 0.0017300. Hence is obtained 

log P'= 9.8907512 
and, therefore, 

X= +0.0014736 
The chief results of the second 



log $'=9.5712864, 

Y= +0.0094574 
!5, in which we put 



a; = log P = 9.8907512 
a^ = log $z= 9.5712864 



are the following ; 
w + a . . 
log ^ c sin w 
z . . . . 
lose / . . . 



20° 8' 0".87 
0.0373071 

21 12 6 .09 
0.3507110 



C 195° 16' 59".90 

r 196 52 40 .63 

logr .... 0.3630642 

loff/' . . . . 0.3369708 



222 



DETERMINATION OF AN ORBIT FROM 



[Book H. 



i{u''-\-u). . . 267" 6'10''.75 2/ 22° 32' 8".69 

^u" — u). . . — 43 39 4.00 2/ 13 154.65 

2/' .... . 9 30 14 .38 
The difference 0."34, between 2/' and 2/ -|- 2/'" is to be so distributed, as to 
make 2/= 13° 1' 54".45, 2f' == 9° 30' 14".24. 

If it is thought worth while to recompute here the corrections of the times, 
there will be found for the first observation, 0.009169, for the second, 0.008742, 
for the third, 0.009236, and thus the corrected times, November 5.564905, Novem- 
ber 36.466293, November 76.340280. Hence we have 

log^ 9.8362703 I log?/' 0.0017413 

log^ 9.7255594 logP' 9.8907268 

log?; 0.0031790 | log ^' 9.5710593 

Accordingly, the results from the second hypothesis are 

X = — 0.0000244, r== — 0.0002271. 
Finally, in the third hypothesis, in which we put 
a; = log P = 9.8907268 
y = log ^ = 9.5710593 
the chief results of the calculation are as follows : — 
(o-^a . . . . 20° 8' 1".62 
log Qc sin (o . . 0.0370857 

3 21 12 4 .60 

log/ 0.3507191 

C 195 16 54 .08 

r 196 52 44 .45 

logr 0.3630960 

The difference 0".38 will be here distributed in such a manner as to make 
2f= 13° 1' 57".20, 2/" = 9° 30' 10".47.* 



logr" . . 


0.3369536 


Htc^' + u). 


. 267 5 53 .09 


Hic"-u). 


.—43 39 4.19 


2/' . . . 


. 22 32 7 .67 


2/ . . . 


. 13 1 57 .42 


2/" . . . 


9 30 10 .63 



* This somewhat increased difference, nearly equal in all the hypotheses, has arisen chiefly from 
this, that (T had been got too little by almost two hundredths of a second, and the logaritlmi of b too 
great by several units. 



Sect. 1.] three complete observations. 223 

Since the differences of all these numbers from those which the second 
hypothesis furnished are very small, it may be safely concluded that the third 
hypothesis requires no further correction, and, therefore, that a new hypothesis 
would be superfluous. Wherefore, it will now be proper to proceed to the calcu- 
lation of the elements from 2/', ^', r, / ' : and since the processes comprised in 
this calculation have been most fully explained above, it will be sufficient to add 
here the resulting elements, for the benefit of those who may wish to perform the 
computation themselves : — 

Eight ascension of the ascending node on the equator .... 158° 40' 38".93 

Inclination of the orbit to the equator 11 42 49 .13 

Distance of the perihelion from the ascending node 323 14 56 .92 

Mean anomaly for the epoch 1806 335 4 13 .05 

Mean daily (sidereal) motion 770".2662 

Angle of eccentricity, (p 14 9 3 .91 

Logarithm of the semi-axis major . 0.4422438 

158. 

The two preceding examples have not yet furnished occasion for using the 
method of article 120 : for the successive hypotheses converged so rapidly that 
we might have stopped at the second, and the third scarcely difiered by a sensible 
amount from the truth. We shall always enjoy this advantage, and be able to do 
without the fourth hypothesis, when the heliocentric motion is not great and the 
three radii vectores are not too unequal, particularly if, in addition to this, the 
intervals of the times differ from each other but little. But the further the con- 
ditions of the problem depart from these, the more will the first assumed values 
of F and Q differ from the correct ones, and the less rapidly will the subsequent 
values converge to the truth. In such a case the first three hypotheses are to 
be completed in the manner shown in the two preceding examples, (with this 
difference only, that the elements themselves are not to be computed in the third 
hypothesis, but, exactly as in the first and second hypotheses, the quantities r], rf^ 
P', Q', X, Y) ; but then, the last values of P', Q' are no longer to be taken as 



224 



DETERMINATION OF AN ORBIT FROM 



[B( 



n. 



the new values of the quantities P, Q in the new hypothesis, but these are to 
be derived from the combination of the first three hypotheses, agreeably to the 
method of article 120. It will then very rarely be requisite to proceed to the 
fifth hypothesis, according to the precepts of article 121. We will now explain 
these calculations further by an example, from which it will appear how far our 
method extends. 

159. 

For the third example we select the following observations of Ceres, the first 
of which has been made by Olbers, at Bremen, the second by Haeding, at Got- 
tingen, and the third by Bessel, at Lilienthal. 



Mean time of place of observation. 


Bight Ascension. 


North declination. 


1805, Sept. 5^3* 8^54' 

1806, .Jan. 17 10 58 51 
1806, May 23 10 23 53 


95° 59' 25" 
101 18 40.6 
121 56 7 


22° 21' 25" 
30 21 22.3 
28 2 45 



As the methods by which the parallax and aberration are taken account of, 
when the distances from the earth are regarded as wholly unknown, have already 
been sufficiently explained in the two preceding examples, we shall dispense 
with this unnecessary increase of labor in this third example, and with that 
object will take the approxunate distances from von Zach's Monaillclie Correr 
spondenz, Vol. XL, p. 284, in order to free the observations from the effects of 
parallax and aberration. The following table shows these distances, together 
with the reductions derived from them : — 



Distance of Ceres from the earth . . . 
Time in which the light reaches the earth 

Eeduced time of observation 

Sidereal time in degrees 

Parallax in right ascension 

Parallax in declination 



2.899 


1.638 


2.964 


23'«49' 


13'"28-' 


24'" 21' 


2H5'" 5' 


10H5'"23^ 


9"59'"32 


355° 55' 


97° 59' 


210° 41 


+ 1".90 


-f 0".22 


— 1".97 


— 2.08 


— 1.90 


— 2.04 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



225 



Accordingly; the data of the problem, after being freed from parallax and 
aberration, and after the times have been reduced to the meridian of Paris, are as 
follows : — 



Times of the observations. 


Eight Ascension. 


DecUnation. 


1805, Sept. 5, 12^ 19™ 14' 

1806, Jan. 17, 10 15 2 
1806, May 23, 9 33 18 


95° 59' 23".10 
101 18 40.38 
121 56 8.97 


22° 21' 27".08 
30 21 24.20 
28 2 47.04 



From these right ascensions and declinations have been deduced the longi- 
tudes and latitudes, using for the obliquity of the ecliptic 23° 27' 55''.90, 23° 2T 
54".59, 23° 27' 63".27 ; the longitudes have been afterwards freed from nutation, 
which was for the respective times -|- 17".31, -|- 17''.88, -(- 18".00, and next re- 
duced to the beginning of the year 1806, by applying the precession -\- 15''.98, 
— 2''.39, — 19".68. Lastly, the places of the sun for the reduced times have 
been taken from the tables, in which the nutation has been omitted in the longi- 
tudes, but the precession has been added in the same way as to the longitudes of 
Ceres. The latitude of the sun has been wholly neglected. In this manner have 
resulted the following numbers to be used in the calculation : — 



Times, 1805, September 
or, a', a" 

/5,i3',r- ..... 

l,V,l" 

log R, log R, log R' . 



5.51336 

95° 32' 18".56 

— 59 34 .06 

342 54 56 .00 

0.0031514 



139.42711 

99° 49' 5".87 

4-7 16 36 .80 

117 12 43 .25 

9.9929861 



265.39813 
118° 5'28".85 
+ 7 38 49 .39 
241 58 50 .71 

0.0056974 



The preliminary computations explained in articles 136-140 furnish the fol- 
io win o; : — 



r, r , r 



AD, AD', AD" 
A'D, A'D', AD' 



358°55'28".09 
112 37 9 .66 

15 32 41 .40 
138 45 4 .60 

29 18 8 .21 
29 



156°52'11".49 

18 48 39 .81 

252 42 19 .14 

6 26 41 .10 

170 32 59 .08 



170°48'44".79 
123 32 52 .13 
136 2 22 .38 
358 5 57 .00 
156 6 25 .25 



226 



DETERMINATION OF AN ORBIT FROM 



[Book IL 



log e = 0.8568244 
log K = 0.1611012 
logj'/'= 9.9770819 ?z 
log 2. = 9.9164090 w 
loo;r= 9.7320127 w 



a = 8° 52' 4".05 
log a = 0.1840193 n,a= — 1.5276340 
log i = 0.0040987 
log c= 2.0066735 
f?= 117.50873 

The interval of time between the first and second observations is 133.91375 
days, between the second and third, 125.97102 : hence 

log ^ = 0.3358520, log r=: 0.3624066, log -^'=0.0265546, log ^r = 0.6982586. 

We now exhibit in the following table the principal results of the first three 
hypotheses : — 





I 


n. 


ni. 


logP^a; 


0.0265546 


0.0256968 


0.0256275 


logQ = ?/ 


0.6982586 


0.7390190 


0.7481055 


w-fa 


7°15'13".523 


7°14'47".139 


7°14'45".071 


log ^c sin to 


1.1546650 ?z 


1.1973925 ;2 


1.2066327 ^z 


e 


7 3 59 .018 


7 2 32 .870 


7 2 16 .900 


log/ 


0.4114726 


0.4129371 


0.4132107 


c 


160 10 46 .74 


160 20 7 .82 


160 22 9 .42 


r 


262 6 1 .03 


262 12 18 .26 


262 14 19 .49 


logr 


0.4323934 


0.4291773 


0.4284841 


log r^ 


0.4094712 


0.4071975 


0.4064697 


*K+^0 


262 55 23 .22 


262 57 6 .83 


262 57 31 .17 


n«"— "/ 


273 28 50 .95 


273 29 15 .06 


273 29 19 .56 


2/ 


62 34 28 .40 


62 49 56 .60 


62 53 57 .06 


2/ 


31 8 30 .03 


31 15 59 .09 


31 18 13 .83 


2r 


31 25 58 .43 


31 33 57 .32 


31 35 43 .32 


logrj 


0.0202496 


0.0203158 


0.0203494 


logV' 


0.0211074 


0.0212429 


0.0212751 


logP' 


0.0256968 


0.0256275 


0.0256289 


log Q' 


0.7390190 


0.7481055 


0.7502337 


X 


— 0.0008578 


— 0.0000693 


-f 0.0000014 


Y 


4-0.0407604 


+ 0.0090865 


4- 0.0021282 



Sect. 1.] 



THREE COMPLETE OBSERVATIONS. 



227 



If we designate the three values of X by A, A', A''; the three values of Y by 
B, B\ B"; the quotients arising from the division of the quantities A'B" — A!'B', 
A^'B — AB", AB' — AB, by. the sum of these quantities, by Ji, Jc\ k", respectively, 
so that we have ^-j-^'-|-£'=: 1 ; and, finally, the values of log P' and log Q' in the 
third hj^othesis, by M and N, (which would become new values of x and p if it 
should be expedient to derive the fourth hypothesis from the third, as the third 
had been derived from the second) : it is easily ascertained from the formulas of 
article 120, that the corrected value of x is M — Jc [A -\- A') — k'A', and the cor- 
rected value of y, iV^ — k [B' -f- B") — k'B". The calculation being made, the 
former becomes 0.0256331, the latter, 0.7509143. Upon these corrected values 
we construct the fourth hypothesis^ the chief results of which are the following : — 



CO -{- (7 . . 
log ^csinw 
. . . . 

log / . . . 

^ . . . . 



7°14'45".247 

1.2094284 n 
7 2 12 .736 
0.4132817 
160 22 45 .38 



log /' 

h{%l' — u) 
2/ . . 
2/ . . 
2/' • . 



0.4062033 

262°5r38".78 

273 29 20 .73 

62 55 16 .64 

31 19 1 .49 

31 36 15 .20 



r 262 15 3 .90 

^ logr 0.4282792 

The difference between 2/' and 2/-|- 2/" proves to be 0''.05, which we shall 
distribute in such a manner as to make 2/= 31° 19' r.47, 2/'= 31° 36' 15".17. 
If now the elements are determined from the two extreme places, the following 
values result : — 

True anomaly for the first place 289° 7'39".75 

True anomaly for the third place 352 2 56 .39 

Mean anomaly for the first place 297 41 35 .65 

Mean anomaly for the third place 353 15 22 .49 

Mean daily sidereal motion 769^.6755 

Mean anomaly for the beginning of the year 1806 . . 322 35 52 .51 

Angle of eccentricity, 9 4 37 57 .78 

Logarithm of the semi-axis major 0.4424661 

By computing from these elements the heliocentric place for the time of the 



228 DETERMINATION OF AN ORBIT FROM [BoOK 11. 

middle observation, the mean anomaly is found to be 326° 19' 25".72, the loga- 
rithm of the radius vector, 0.4132825, the true anomaly, 320° 43' 54".87 : this last 
should differ from the true anomaly for the first place by the quantity 2/", or 
from the true anomaly for the third place by the quantity 2/, and should, there- 
fore, be 320° 43' 54".92, as also the logarithm of the radius vector, 0.4132817 : 
the difference 0".05 in the true anomaly, and of eight units in the logarithm, is 
to be considered as of no consequence. 

If the fourth hypothesis should be conducted to the end in the same way as 
the three preceding, we would have X= 0, Y= 0.0000168, whence the follow- 
ing corrected values of x and y would be obtained, 

X = log P = 0.0256331, (the same as in the fourth hypothesis,) 

^ = log ^ = 0.7508917. 
If the fifth hypothesis should be constructed on these values, the solution would 
reach the utmost precision the tables allow: but the resulting elements would 
not differ sensibly from those which the fourth hypothesis has furnished. 

Nothing remains now, to obtain the complete elements, except that the posi- 
tion of the plane of the orbit should be computed. By the precepts of article 
149 we have 

From the first place. From the third place. 

ff 354° 9' 44".22 / . . . . 57° 5' 0".91 

h 261 56 6 .94 r .... 161 1 .61 

i 10 37 33 .02 10 37 33 .00 

9, 80 58 49 .06 80 58 49 .10 

Distance of the perihelion 



,, 65 2 4 .47 65 2 4 .52 

from the ascendmg node 

Longitude of the perihelion 146 53 .53 146 53 .62 

The mean being taken, we shall put i= 10° 37' 33".01, 9, = 80° 58' 49".08, the 
longitude of the perihelion = 146° 0' 53".57. Lastly, the mean longitude for 
the beginning of the year 1806 will be 108° 36' 46".08. 



Sect. 1.] three complete observations. 229 

160. 

In the exposition of the method to which the preceding investigations have 
been devoted, we have come upon certain special cases to which it did not apply, 
at least not in the form in which it has been exhibited by us. We have seen 
that this defect occurs first, when any one of the three geocentric places coincides 
either with the corresponding heliocentric place of the earth, or with the oppo- 
site point (the last case can evidently only happen when the heavenly body 
passes between the sun and earth) : second, when the first geocentric place of the 
heavenly body coincides with the third j third, when aU three of the geocentric 
places together with the second heliocentric place of the earth are situated in the 
same great circle. 

In the first case the position of one of the great circles AB, AB', A.'B", and in 
the second and third the place of the point B^, wUl remain indeterminate. In 
these cases, therefore, the methods before explained, by means of which we have 
shown how to determine the heliocentric from the geocentric places, if the quan- 
tities P, Q, are regarded as known, lose their efl&cacy : but an essential distinction 
is here to be noted, which is, that in the first case the defect will be attributable 
to the method alone, but in the second and third cases to the nature of the prob- 
lem; in the first case, accordingly, that determination can undoubtedly be effected 
if the method is suitably altered, but ia the second and third it will be absolutely 
impossible, and the heliocentric places will remain indeterminate. It will not be 
uninteresting to develop these relations in a few words : but it would be out of 
place to go through all that belongs to this subject, the more so, because in all 
these special cases the exact determination of the orbit is impossible where it 
would be greatly afiected by the smallest errors of observation. The same defect 
will also exist when the observations resemble, not exactly indeed, but nearly, 
any one of these cases ; for which reason, in selecting observations this is to be 
recollected, and properly guarded against, that no place be chosen where the 
heavenly body is at the same time in the vicinity of the node and of opposition 
or conjunction, nor such observations as where the heavenly body has nearly re- 
turned in the last to the geocentric place of the first observation, nor, finally, such 



230 DETEEMINATION OF AN ORBIT FROM [BoOK 11. 

as where the great circle drawn from the middle heliocentric place of the earth to 
the middle geocentric place of the heavenly body makes a very acute angle with 
the direction of the geocentric motion, and nearly passes through the first and 
third places. 

161. 

We will make three subdivisions of the first case. 

I If the point B coincides with A or with the opposite point, 8 will be equal 

to zero, or to 180° ; y, t', t" and the points D', D", will be indeterminate ; on the 

other hand, y', y", e and the points B, B% will, be determinate ; the point C will 

necessarily coincide with A. By a course of reasoning similar to that pursued in 

article 140, the following equation will be easily obtained : — 

^ , sin (z — a) R sin d' sin {A!'D — 8") „ 

^ ~ ^^ sin ^ i2" sin ,5" sin {A!D — 5' + <t) '^ ' 

It will be proper, therefore, to apply in this place all which has been explained in 
articles 141, 142, if, only, we put a = 0, and h is determined by equation 12, 
article 140, and the quantities z, r', — , — ^, will be computed in the same manner 
as before. Now as soon as z and the position of the point C have become 
known, it will be possible to assign the position of the great circle CC, its inter- 
section with the great circle A'B", that is the point G", and hence the arcs CC, 
CC", CC", or 2/", 2f', 2/. Lastly, from these will be had 

n'r' sin 2 f „ ?iVsin 2 f" 
n sin 2/' ' vl' sin 2/' ' 

n. Every thing we have just said can be applied to that case in which B" 
coincides with A!.' or with the opposite point, if, only, all that refers to the first 
place is exchanged with what relates to the third place. 

HE. But it is necessary to treat a little difierently the case in which B' coin- 
cides with A. or with the opposite point. There the point C will coincide with 
A. ', Y, e, t" and the points D,D",B^, will be indeterminate: on the other hand, 
the intersection of the great circle BB" with the ecliptic,f the longitude of which 

t More generally, with the great circle AA' : but for the sake of brevity we are now considering 
that case only where the ecliptic is taken as the fundamental plane. 



Sect. 1.] three complete observations. 231 

maybe put equal to l' -\- tt, may be determined. By reasonings analogous to 
those which have been developed in article 140, will be obtained the equation 

S sin d sin (A" ly — 8") 






E" sin d" sin (AD' — d) ' R" sin {I" -- V — n) ' 

Let us designate the coefficient of n, which agrees with a, article 140, by the 
same symbol a, and the coefficient of n'r by /i : a may be here also determined 
by the formula 

Rsin{l'-[-7t — l) 

^~ R"sin{l" — l'—ny 

We have, therefore, 

= aJ^-f-/3^^y-{-w", 

which equation combined with these, 
produces 

whence we shall be able to get /, unless, indeed, we should have /3 = 0, in which 
case nothing else would follow from it except P = — a. Further, although we 
might not have (? = (when we should have the third case to be considered in 
the following article), still (3 will always be a very small quantity, and therefore 
P wi|l necessarily differ but little from — a : hence it is evident that the deter- 
mination of the coefficient 

is very uncertain, and that /, therefore, is not determinable with any accuracy. 
Moreover, we shall have 

wV t^i^ ^ P+a 

~^~ ~f~' n" ~~ ~JP~' 

after this, the following equations will be easily developed in the same manner as 
in article 143, 

r sm C = — ^-r sm (r — I), 



232 DETERMINATION OF AN ORBIT FROM [BoOK 11. 

// • ^11 n'/sinr . ,,, ,, 

r" ^Vixi,' ^— ^^r^mx{l —I), 

r sin U: — AD') = r"P '^^ sin (l," — A'D'), 

from the combination of which with equations VIII. and IX. of article 143, the 
quantities r, t? '>'"■) Q" can be determined. The remaining processes of the calcula- 
tion will agree with those previously described. 

162. 

In the second case, where B" coincides with B, D' will also coincide with them 
or with the opposite point. Accordingly, we shall have AD' — d and A'l/ — d" 
either equal to or 180° : whence, from the equations of article 143, we obtain 

sms' R sin (5 



= + 



sin i' H' sin 5" 



n" — sin a" sin (z -|- A' I/' — «5') ' 
R sin d sin e" sin {z -\- AD" — d') = PR" sin d" sin e sin (s + A'D — d'). 

Hence it is evident that z is determinable by P alone, independently of Q, (un- 
less it should happen that AD" = AD, or = AD + 180°, when we should have 
the third case) : s being found, / will also be known, and hence, by means of 
the values of the quantities 



and, lastly, from this also 



n'r' nV ^ n ■, n" 

— , —ff, also -r and -7; 



^ = 2fc + 'i;-l)/l 



Evidently, therefore, P and Q cannot be considered as data independent of each 
other, but they will either supply a single datum only, or inconsistent data. The 
positions of the points C, 0" will in this case remain arbitrary, if they are only 
taken in the same great circle as 0' . 

In the third case, where A, B, B', B", lie in the same great circle, D and D" will 
coincide with the points B", B, respectively, or with the opposite points : hence is 



Sect. 1.] THREE COMPLETE OBSERVATIONS. 233 

obtained from the combination of equations VII., VIII., IX., article 143, 

p ^ sin 5 sin &" B sin (I' — I) 

i^" sin fi" sine R"sm{l" — V) ' 

In this case, therefore, the value of P is had from the data of the problem, and, 
therefore, the positions of the points 0, C, G", will remain indeterminate. 

163. 

The method which we have fully explained from article 136 forwards, is prin- 
cipally suited to the first determination of a wholly unknown orbit : still it is em- 
ployed with equally great success, where the object is the correction of an orbit 
already approximately known by means of three observations however distant 
from each other. But in such a case it will be convenient to change some things. 
When, for example, the observations embrace a very great heliocentric motion, it 
will no longer be admissible to consider -j and & ^" as approximate values of the 
quantities P, Q : but much more exact values will be obtained from the very 
nearly known elements. Accordingly, the heliocentric places in orbit for the 
three times of observation will be computed roughly by means of these elements, 
whence, denoting the true anomalies by v, v' , v", the radii vectores by r, /, r'\ the 
semi-parameter by j», the following approximate values will result : — 

p r sin {v' — v) ^ 4 r'^ sin \ (v' — v) sin \ {v" — v') 

/' sin (vl' — w') ' ^ pcos^ (v" — v) 

With these, therefore, the first hypothesis will be constructed, and with them, a 
little changed at pleasure, the second and third : it would be of no advantage 
to adopt F' and Q' for the new values, since we are uo longer at liberty to sup- 
pose that these values come out more exact. For this reason all three of the 
hypotheses can be most conveniently despatched at the same time: the fourth will 
then be formed according to the precepts of article 120. FinaiUy, we shall not 
object, if any person thinks that some one of the ten methods explained in arti- 
cles 124-129 is, if not more, at least almost equally expeditious, and prefers to 
use it, 

30 



SECOND SECTION. 



DETERSIINATION OF AN ORBIT FROM FOUR OBSERVATIONS, OF WHICH TWO 
ONLY ARE COMPLETE. 



164. 

We have already, in the beginning of the second hook (article 115), stated 
that the use of the problem treated at length in the preceding section is lim- 
ited to those orbits of which the inclination is neither nothing, nor very small, 
and that the determination of orbits slightly inclined must necessarily be based 
on four observations. But four complete observations, since they are equivalent 
to eight equations, and the number of the unknown quantities amounts only to 
six, would render the problem more than determinate : on which account it wiU 
be necessary to set aside from two observations the latitudes (or declinations), 
that the remaining data may be exactly satisfied. Thus a problem arises to 
which this section will be devoted : but the solution we shall here give will ex- 
tend not only to orbits slightly inclined, bul; can be applied also with equal suc- 
cess to orbits, of any inclination however great. Here also, as in the problem of 
the preceding section, it is necessary to separate the case, in which the approxi- 
mate dimensions of the orbit are already known, from the first determination 
of a wholly unknown orbit : we will begin with the former. 

165. 

The simplest method of adjusting a known orbit to satisfj^ four observations 
appears to be this. Let x^ y, be the approximate distances of the heavenly body 
from the earth in two complete observations : by means of these the correspond- 
ing heliocentric places may be computed, and hence the elements; after this, 
(234) 



Sect. 2.] determination of an orbit. 235 

from these elements the geocentric longitudes or right ascensions for the two 
remaining observations may be computed. If these happen to agree with the 
observations, the elements will require no further correction: but if not, the 
differences X, Y, will be noted, and the same calculation will be repeated twice, 
the values of x, y being a little changed. Thus will be obtained three systems 
of values of the quantities x, y, and of the differences X, Y, whence, according 
to the precepts of article 120, will be obtained the corrected values of the quan- 
tities :?;, ^, to which will correspond the values X== 0, Jr=: 0. From a similar 
calculation based on this fourth system elements will be found, by which all four 
observations will be correctly represented. 

If it is in your power to choose, it will be best to retain those observations 
complete from which the situation of the orbit can be determined with the great- 
est precision, therefore the two extreme observations, when they embrac-e a helio- 
centric motion of 90° or less. But if they do not possess equal accuracy, you 
will set aside the latitudes or declinations of those you may suspect to be the 
less accurate. 

166. 

Such places will necessarily be used for the first determination of an entirely 
unknown orbit from four observations, as include a hehocentric motion not too 
great ; for otherwise we should be without the aids for forming conveniently the 
first approximation. The method which we shall give directly admits of such 
extensive application, that observations comprehending a heliocentric motion of 
30° or 40° may be used without hesitation, provided, only, the distances from the 
sun are not too unequal : where there is a choice, it will be best to take the 
intervals of the times between the first and second, the second and third, the 
third and fourth but little removed from equality. But it will not be necessary 
to be very particular in regard to this, as ,the annexed example wUl show, in 
which the intervals of the times are 48, 55, and 59 days, and the heliocentric 
motion more than 50°. 

Moreover, our solution requires that the second and third observations be 
complete, and, therefore, the latitudes or declinations in the extreme observations 



236 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BoOK II. 

are neglected. We have, indeed, shown above that, for the sake of accuracy, it is 
generally better that the elements be adapted to two extreme complete observa- 
tions, and to the longitudes or right ascensions of the intermediate ones ; never- 
theless, we shall not regret having lost this advantage in the first detennination 
of the orbit, because the most rapid approximation is by far the most important, 
and the loss, which affects chiefly the longitude of the node and the inclina- 
tion of the orbit, and hardly, in a sensible degree, the other elements, can after- 
wards easily be remedied. 

We will, for the sake of brevity, so arrange the explanation of the method, 
as to refer all the places to the ecliptic, and, therefore, we will suppose four longi- 
tudes and two latitudes to be given : but yet, as we take into account the latitude 
of the earth in our formulas, they can easily be transferred to the case in which 
the equator is taken as the fundamental plane, provided that right ascensions and 
declinations are substituted in the place of longitudes and latitudes. 

Finally, all that we have stated in the preceding section with respect to nutar 
tion, precession, and parallax, and also aberration, applies as well here : unless, 
therefore, the approximate distances from the earth are otherwise known, so that 
method I., article 118, can be employed, the observed places will in the beginning 
be freed from the aberration of the fixed stars only, and the times will be cor- 
rected as soon as the approximate determination of the distances is obtained in 
the course of the calculation, as will appear more clearly in the sequel. 

167. 

We preface the explanation of the solution with a list of the principal sym- 
bols. We will make 

i, i', f, f, the times of the four observations, 

a, a, a", a'", the geocentric longitudes of the heavenly body, 

/i, /?', /5", (i"\ their latitudes, 

r, r', /', r", the distances from the sun, 

q, ()', q", q'", the distances from the earth, 

I, I', I", I'", the heliocentric longitudes of the earth, 



Sect. 2.] OF which two only are complete. 237 

B, B', B", B'", the heliocentric latitudes of the earth, 
R, R', R'\ R'", the distances of the earth from the sun, 

(wOl), (w 12), («23), (w 02), (« 13), the duplicate areas of the triangles which 
are contained between the sun and the first and second places of the heavenly 
body, the second and third, the third and fourth, the first and third, the second 
and fourth respectively; [ri 01), [r] 12), (tj 23) the quotients arising from the 
division of the areas h {n 01), 2 {n 12), ^ [n 23), by the areas of the correspond- 
ing sectors ; 

(n 01) ' ~ (n 23) ' 

V, v', v", v'", the longitudes of the heavenly body in orbit reckoned from an arbi- 
trary point. Lastly, for the second and third observations, we will denote the 
heliocentric places of the earth in the celestial sphere by JL', JL", the geocentric 
places of the heavenly body by B\ B", and its heliocentric places by C, C" . 

These things being understood^ the first step will consist, exactly as in the 
problem of the preceding section (article 136), in the determination of the posi- 
tions of the great circles A!. C'B', A' C"B", the inclinations of which to the eclip- 
tic we denote by /', y": the determination of the arcs AB'=id', A'B"z=d" will be 
connected at the same time with this calculation. Hence we shall evidently have 
r' = sl {q'q' + 2 q'R' cos d' -\- R'R') 
r"= si {q'Y + 2 Q"Ji" cos 6" + R"R''), 
or by putting 9' + R' cos d' = x', q" + R" cos d" = a!', R! sin d' — a', R" sin d" = a", 

r' = \J (// + a'a') 
r"=sl{x"^'-\-a;'a"). 

168. 

By combining equations 1 and 2, article 112, the following equations in sjm.- 
bols of the present discussion are produced : — 

= («12)i?cos^sin(/— a) — (?^02)(9'cos/5'sin(a'— a)-f i2'cos^'sin(r— a)) 
4- {n 01) (y" cos {i" sin {a" — «) + R" cos B" sin {I" — a)), 



238 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BuOK II. 

= {n 23) (^/ COS /5' sin (a'"— «') + B' cos B' sin (a'"— /')) 
— {n 13) (9" cos /?"sin (a"'— a") + i^'' cos B'' sin (a'"— T)) 
+ {n 12) i?"' cos B'" sin (a'" — r"). 
These equations, bj putting 

i2' cos 5' sin (Z' — a) -r,/ .>/ 7/ 

n, . , , r-^ B COS a =J, 

cos p sin (a — a) ' 

jB"cos5"sm(a'"— Z") „„ ^„ ,„ 
-coT^'sMi?^^-^ ««^^ =^' 

ig'cos^sin(«--0 _ ^ ^^^ ^, _ ^, 
cos (3 sm (a — a ) ' 

jy- cos ^;- sin (I" -a) _ ^. ^^^ ^. ^ ^. 

cosp sin (« — a) 

i? cos B sin (Z — a) . 

cos (3" sin (a" — «) ' 

Jg^"cos^"Mn(«^^^ — r^O _ . ,„ 
cos/3'sin («'" — «') ~ » 

cos §' sin (a' — a) / 

cos ^' sin (a" — a) ^' 

cos|y^sin(«'^^ — «'0 „ 

cos |3'sin («'" — a') ^ ' 

and all the reductions being properly made, are transformed into the following: — 
/(l-f-PO(:c^ + y) 



14 



Q' 



+ x" + ^i^, 

iH — -3 

(a/'s/'-\-a"a"f 

or, by putting besides, 

_ k'— rF"= c", ^" (1 + P") = /', 
into these, 

L y' = o-+ ^' <-" + ''> 



(xV -}-«'«') 



Sect. 2.] OF wmcH two only ake complete. 239 



(a/V'+a"a")* 



With the aid of these two equations / and x' can be determined from d, h', c', d', 
Q'. a", h", c", d", Q". If, indeed, / or x" should be eliminated from them, we should 
obtain an equation of a very high order : but still the values of the unknown 
quantities x', x", will be deduced quickly enough from these equations by indi- 
rect methods without any change of form. Generally approximate values of 
the unknown quantities result if, at first, Q' and Q" are neglected ; thus : — 

^ _ </^ -f d" {W -I- c') -f d'd"b' 
^— 1—d'd" ' 



„_ c'-\- d' {V + </0 + d'd"V' 
^ — 1-d'd" 



But as soon as the approximate value of either unknown quantity is obtained, 
values exactly satisfying the equations will be very easily found. Let, for ex- 
ample, t' be an approximate value of x, which being substituted in equation I., 
there results x" =-1" ', in the same manner from x" = ^" being substituted in 
equation II., we may have x' =^ X'; the same processes may be repeated by sub- 
stituting for X in I., another value h,' -j- v', which may give x" = ^" -\- v" ; this 
value being substituted in II., may give x' = X' -\- N\ Thereupon the corrected 
value of X will be 



5 "T iV'—y — i^'_/ 



and the corrected value of x", 






If it is thought worth while, the same processes will be repeated with the cor- 
rected value of x' and another one sHghtly changed, until values of x', x" satisfy- 
ing the equations I., 11. exactly, shall have been found. Besides, means will not 
be wanting even to the moderately versed analyst of abridging the calculation. 

In these operations the irrational quantities (a;V-|- aV)^', (a;' V-f-«"fif")^ ^^® 
conveniently calculated by introducing the arcs /, z", of which the tangents are 



240 DETERMINATION OF AN ORBIT FROM FOUK OBSERVATIONS, [BoOK II. 

respectively ^, -7,, whence come 

v/(^V-[-«V)=/ = ^ = ^, 

' ^ ' ' sm/ cosz" 

I / r/ r/ \ r/ n\ it O, X 

\J ix X 4-aa ) = r =:-^ — 7, = j,. 

' ^ ' ' sin 2" cosz' 

These auxiliary arcs, which must be taken between 0° and 180°, in order that 
/, r", may come out positive will, manifestly, be identical with the arcs C'B', G"B", 
whence it is evident that in this way not only / and r", but also the situation of 
the points C, C", are known. 

This determination of the quantities x', x" requires a', a", V , h", c\ c", d', d'\ (^, 
Q" to be known, the first four of which quantities are, in fact, had from the data 
of the problem, but the four following depend on P', P". Now the quantities 
P', P", Q', Q", cannot yet be exactly determined ; but yet, since 

^- ^ -t'-t (rjuy 

TV P" ^'~^' C'?^^) 

V. Q =hJcJc{if — t){t — f) ^, (^ 01) {t] 12) cos 1 (v' — v) cos i (t/' — v) cos \ (i-" — v') ' 
VI. Q" = i kic {f— /) (r — f) y^ P2) (v?23) cos\{v"—v') cosi (z;'"-«') cos\^{v"' — v")' 

the approximate values are immediately at hand, 
p' ^ ^ jp" ^' — ^ 

q = hJck it' — t) [f — If), Q" =HJc {t" — t') {f — r ), 

on which the first calculation will be based. 

169. 

The calculation of the preceding article being completed, it will be necessary 
first to determine the arc C C". Which may be most conveniently done, if, as 
in article 137, the intersection D of the great circles AfC'B', Al' C"B", and their 
mutual inclination e shall have been previously determined: after this, will be 
found from £, CD = s'-\- B'D, and 0"J) = z" -\- B"D, by the same formulas 



Sect. 2.] or which two only are complete. 241 

which we have given in article 144, not only C G" ^v" — /, but also the angles 
{%{, u",) at which the great circles AB', A'B", cut the great circle C C". 

After the arc v" — v' has been found, v' — y, and r will be obtained from a 
combination of the equations 

r sm [y — v)^= ^^t , 



r sin {v' — v -^ v" — /) — , p, „, 

-I I V 



1+. 

and ia the same manner, /" and v'" — v" from a combination of these: — 

in • r III II \ 1^ sin (v' — v') 

r sm (y — V ) = —^ , 

r'" sin {v'" - v" + v" - v') = ^^- ^'^-<^'-f) . 

1 + J 
All the numbers found in this manner would be accurate if we could set out in 
the beginning from true values of P', Q^, P", Q'^ : and then the position of the 
plane of the orbit might be determined in the same manner as in article 149, 
either from A'C, u' and /, or from A' C", u" and y" -, and the dimensions of the 
orbit either from /, /', t', f, and v" — v', or, which is more exact, from r, r", t, 
f, v'" — V. But in the first calculation we will pass by all these things, and will 
direct our attention chiefly to obtaining the most approximate values of P' , P" . 
Q', Q". We shall reach this end, if by the method explained in 88 and the fol- 
lowing articles, 

from r, /, v' — v,f — t we obtain {rj 01) 

« /^r",v" — v',f — t' « (7jl2) 

« r"yy—v",f'—f •'' (7j23). 
We shall substitute these quantities, and also the values of r, /, /', /", cos i {v' — v), 
etc., in formulas III.- VI., whence the values of P', Q', P", Q" will result much 
more exact than those on which the first hypothesis had been constructed. With 
these, accordingly, the second hypothesis will be formed, which, if it is carried to 
a conclusion exactly in the same manner as the first, will furnish much more 
exact values of I*', Q', P", Q', and thus lead to the third hypothesis. These 
processes will continue to be repeated, until the values of P', Q, P", Q" seem to 

31 



242 DETERMINATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BoOK II. 

require no further correction, how to judge correctly of which, frequent practice 
will in time show. When the heliocentric motion is small, the first hypothesis 
generally supplies those values with sufficient accuracy : but if the motion in- 
cludes a greater arc, if, moreover, the intervals of the times are very unequal, 
hypotheses several times repeated will be wanted ; but in such a case the first 
hypotheses do not demand great preciseness of calculation. Finally, in the last 
hypothesis, the elements themselves will be determined as we have just indicated. 

170. 

It will be necessary in the first hypothesis to make use of the times t, if, f, f, 
uncorrected, because the distances from the earth cannot yet be computed': as 
soon, however, as the approximate values of the quantities x', x" have become 
known, we shall be able to determine also those distances approximately. But 
yet, since the formulas for q and q'" come out here a little more complicated, it 
will be weU to put off the computation of the correction of the times until the 
values of the distances have become correct enough to render a repetition of the 
work unnecessary. On which account it will be expedient to base this operation 
on those values of the quantities x', x", to which the last hypothesis but one leads, 
so that the last hypothesis may start with corrected values of the times and of 
the quantities P', P'\ Q'y Q". The following are the formulas to be employed 
for this purpose : — 

Vn. q' = x'—EQOBd\ 
Ym. f=xf'—R'coBd", 
IX. Qcos(i = — i? cos -5 cos (a — /) 

-j — — ^, (q' cos (^' cos (a' — a) -\- E' cos B' cos U' — a)) 

— -^(^" cos ^" COS («"—«) 4-^" COS ^" cos (r—«)), 

X. Qsm^ = — JRsmB^ ^^"^-^ ((>' sin (r + R' sin B") 

^'(1 + 7-a) 
— -i (^/' sin (r + B" sin B") , 



Sect. 2.] OF which two only aee complete. 243 

XI. q'" cos r = — R'" cos B'" cos {a'" —I'") 

H ^^'^" rv' (q" cos ir COS {a'" — a") -{- R" cos B" cos (a'" — l")) 



xn. 



— 'mW cos ii' cos {a'" — a) -\- R' cos B' cos (a'" — I'yj, 

r sin /3"' = — R!" sin ^"' -\ ^"^^^, (^'' sin ^" + iT' sin .g'') 

^"(l + ls) 

^,(9'sin/'r-hi?'sin^'). 



The formulas IX.-XII. are derived without difficulty from equations 1, 2, 3, article 
112, if, merely, the symbols there used are properly converted into those we here 
employ. The formulas will evidently come out much more simple if B, B' , B" 
vanish. Not only (), but also /5 will follow from the combination of the formulas 
IX. and X., and, in the same manner, besides r" , also {^"' from XI. and XII. : the 
values of these, compared with the observed latitudes (not entering into the 
calculation), if they have been given, will show with what degree of accuracy 
the extreme latitudes may be represented by elements adapted to the six remain- 
ing data. 

171. 

A suitable example for the illustration of this mvestigation is taken from Vesta, 
which, of all the most recently discovered planets, has the least inclination to 
the ecliptic* We select the following observations made at Bremen, Paris, 
Lilienthal, and Milan, by the illustrious astronomers Olbers, Bouvard, Bessel, and 
Oriani : — 



* Nevertheless this inclination is still great enough to admit of a sufficiently safe and accurate deter- 
mination of the orbit based upon three observations : in fact the first elements which had been derived 
in this way from observations only 19 days distant from each other (see von Zach's Monatliche Gor- 
respondenz, Vol. XV. p. 595), approach nearly to those which were here deduced from four observa- 
tions, removed from each other 162 days. 



244 



DETERlNnNATION OF AN ORBIT FROM FOUR OBSERVATIONS, [BoOK II. 



Mean time of place of observation. 


Riglit Ascension. 


Declination. 


1807, March 30, 12'^ 33"* 17* 
May 17, 8 16 5 
July 11, 10 30 19 
Sept. 8, 7 22 16 


183° 52' 40".8 
178 36 42.3 
189 49 7.7 
212 50 3.4 


11° 54' 27".0 N. 
11 39 46.8 

3 9 10 .IN. 

8 38 17 .OS. 



We find for the same times from the tables of the sun, 





Longitude of the Sun 
from app. Equinox. 


Nutation. 


Distance from 
the Earth. 


Latitude of 
the Sun. 


Apparent obliquity 
of the Ecliptic. 


March 30 
May 17 
July 11 
Sept. 8 


9° 21' 59".5 

55 56 20 .0 

108 34 53 .3 

165 8 57 .1 


: 


hl6.8 
-16.2 
-17.3 
-16.7 


0.9996448 
1.0119789 
1.0165795 
1.0067421 


+ 0".23 

— 0.63 

— 0.46 
+ 0.29 


23° 27' 50".82 
49.83 
49 .19 

23 27 49 .26 



The observed places of the planets have, the apparent obliquity of the ecHp- 
tic being used, been converted into longitudes and latitudes, been freed from 
nutation and aberration of the fixed stars, and, lastly, reduced, the precession 
being subtracted, to the beginning of the year 1807 ; the fictitious places of the 
earth have then been derived from the places of the sun by the precepts of arti- 
cle 72 (in order to take account of the parallax), and the longitudes transferred 
to the same epoch by subtracting the nutation and precession ; finally, the times 
have been counted from the beginning of the year and reduced to the meridian 
of Paris. In this manner have been obtained the following; numbers : — 



i,t',t",r . . 


89.505162 


, 137.344502 


192.419502 


251.288102 


a, a', a", a'". . 


178° 43' 38".87 


174° r30".08 


187°45'42".23 


213°34'15".63 


(i,(r,(r,r^ . 


12 27 6 .16 


10 8 7 .80 


6 47 25 .51 


4 20 21 .63 


I, r, r, r . . 


189 21 33 .71 


235 56 .63 


288 35 20 .32 


345 9 18 .69 


log B,R',R",R"' 


9.9997990 


0.0051376 


0.0071739 


0.0030625 


Hence we deduce 




/=168°32'4r.34, r= 62° 23' 4".8 


8, log a = 9.9526104, 


/ = 173 


5 15 .68, d" 


= 100 45 1 .4 


0, loga'=9 


.9994839, 



Sect. 2.] of which two only ake complete. 245 

y = — 11.009449, x' = — 1.083306, log I = 0.0728800, log / = 9.7139702 n 
b" = — 2.082036, x'' = -1-6.322006, logr'= 0.0798512?z log/*" =9.8387061 

AD= 37°ir5r.50, rD= 89°24'ir.84, £ = 9° 5' 5".48 

B'D = — 26 5 13 .38, B"I) = — 11 20 49 .56. 

These preliminary calculations completed, we enter upon the ^rst hypothesis. 
From the intervals of the times we obtain 

log i{i; — t) = 9.9153666 
logy&(r — = 9.9765359 
logy^(r— = 0.0054651, 
and hence the first approximate values 

log P' = 0.06117, log (1 -f F') = 0.33269, log Q" = 9.59087 
logP"=: 9.97107, log(l -f-P") = 0.28681, log Q"= 9.67997, 
hence, further, 

e' = — 7.68361, log d' = 0.04666 n 
c"= -\- 2.20771, log d"= 0.12552. 
With these values the following solution of equations I., IE., is obtained, after a 
few trials : — 

x' = 2.04856, / = 23° 38' 17", log / = 0.34951 
/'= 1.95745, 0"=:27 2 0, logr"= 0.34194. 

From s', z" and £, we get 

C"C"=«;" — y' = 17° r 5": 
hence v' — v, r, v'" — y", r", will be determinable by the following equations : — 
log r sin {v' — v)= 9.74942, log r sin (/ — y -j- 17° 1' 5") = 0.07600 
log/"sin(y"'—z;")= 9.84729, log/"sin(/"— y''^- 17 7 5") = 0.10733 
whence we derive 

v' — v = 14° 14' 32", log r = 0.35865 
v"'—v"= 18 48 33, logr"'= 0.33887. 

Lastly, is found 

log {n 01) = 0.00426, log {n 12) == 0.00599, log {n 23) = 0.00711, 
and hence the corrected values of P', P", Q', Q\ 



246 DETER]\nNATION OF AN ORBIT FROM FOUR OBSERYATIONS, [BoOK 11. 

log P' = 0.05944, log Q' = 9.60374, 
log P"= 9.97219, log Q"= 9.69581, 
upon which the second hjpoihem will be constructed. The principal results of this 
are as follows : — 

c' = — 7.67820, log ^' = 0.045736 n 
c"= + 2.21061, logc/"= 0.126054 
d = 2.03308, / = 23° 47' 54", log / = 0.346747, 
:?;"= 1.94290, g''= 27 12 25, log /'== 0.339373 
C'C"=^v" — v'=\T 8' 0'' 
v' — v= 14° 21' 36", log r = 0.354687 
v"'—v"=\'^ 50 43, logr"'= 0.334564 

log {n 01) = 0.004359, log (w 12) = 0.006102, log [n 23) = 0.007280. 
Hence result newly corrected values of P' , P", Q', Q", 

log P' = 0.059426, log Q' = 9.604749 
log P" = 9.972249, log Q" = 9.697564, 
from which, if we proceed to the third hypothesis, the follomng numbers result : — 
d =. — 7.67815, log d! = 0.045729 n 
c" = 4- 2.21076, log d"= 0.126082 
x' = 2.03255, z = 23° 48' 14", log / = 0.346653 
^"=1.94235, /'=27 12 49, log r"= 0.339276 
C'0"=v''—v'=l7° 8' 4" 
v' — v= 14° 21' 49", logr =0.354522 
t,'"_z;''=18 51 7, log/"= 0.334290 

log (w 01) = 0.004363, log (^? 12) = 0.006106, log 0? 23) = 0.007290. 
If now the distances from the earth are computed according to the precepts of 
the preceding article, there appears : — • 

9' = 1.5635, 9" =2.1319 

log 9 cos /3 = 0.09876 log q" cos /?'" = 0.42842 

log 9 sin /? = 9.44252 log 9'" sin |3"' = 9.30905 

/? = 12° 26' 40" /5'" = 4° 20' 39" 

log 9 = 0.10909 log 9'" = 0.42967. 



Sect. 2.] 



OF WHICH TWO ONLY ARE COMPLETE. 



247 



Hence are found 





Corrections of the Times. 


Corrected Times. 


I. 


0.007335 


89.497827 


n. 


0.008921 


135.335581 


m. 


0.012165 


192.407337 


IV. 


0.015346 


251.272756 



whence wiU result newly corrected values of the quantities P', JP", Q', Q", 

log P' == 0.059415, log Q' = 9.604782, 
logP''= 9.972253, log Q'' = 9.697687. 
Finally, if the fourth hypothesk is formed with these new values, the following 
numbers are obtained : — 

c' = — 7.678116, log d! = 0.045728 

c"= + 2.210773, log ^"=0.126084 

x' = 2.032473, s' = 23° 48' 16". 7, log / = 0.346638 

a;"= 1.942281, ^'=27 12 51.7, log /'= 0.339263 

v''—v' = 17° 8' 5".l, i(«t''4-w0 = 176° r^0".6, ^u"—u') = ^°SS'2r.e 

v' — v = U 21 51 .9, log r = 0.354503 

v'''—v"=lS 51 9 .5, logr'"= 0.334263 

These numbers differ so little from those which the third hypothesis furnished, 
that we may now safely proceed to the determination of the elements. In 
the first place we get out the position of the plane of the orbit. The inclina- 
tion of the orbit 7° 8' 14".8 is found by the precepts of article 149 from /', iif, 
and ^'^' = ^' — /, also the longitude of the ascending node 103° 16'37".2, the 
argument of the latitude in the second observation 94° 36' 4". 9, and, there- 
fore, the longitude in orbit 197° 52' 42".l ; in the same manner, from 7", it", and 
-A" C" = d" — /', are derived the inclination of the orbit = 7° 8' 14".8, the longi- 
tude of the ascending node 103° 16' 37".5, the argument of the latitude in the 
third observation 111° 44' 9". 7, and therefore the longitude in orbit 215° 0'47".2. 
Hence the longitude in orbit for the first observation will be 183° 30' 50".2, for 
the fourth 233° 51' 56". 7. If now the dimensions of the orbit are determined 
from r — t, r, r'", and v'" — y = 50° 21' 6".5, we shall have. 



248 DETERJiONATION OF AN ORBIT FROM FOUR OBSERVATIONS. [BoOK 11. 

True anomaly for the first place 293°33'43".7 

True anomaly for the fourth place 343 54 50 .2 

Hence the longitude of the perihelion 249 57 6 .5 

Mean anomaly for the first place 302 33 32 .6 

Mean anomaly for the fourth place 346 32 25.2 

Mean daily sidereal motion 9 7 8''. 72 16 

Mean anomaly for the beginning of the year 1807 . 278 13 39 .1 

Mean longitude for the same epoch 168 10 45 .6 

Angle of eccentricity (p 5 2 58 .1 

Logarithm of the semi-axis major 0.372898 

K the geocentric places of the planet are computed from these elements 
for the corrected times t, f, f, f, the four longitudes agree with «, a', a'\ a", and 
the two intermediate latitudes with {^', 1^", to the tenth of a second ; but the 
extreme latitudes come out 12° 26' 43".7 and 4° 20' 40".l. The former in error 
22".4 in defect, the latter 18".5 in excess. But yet, if the inclination of the 
orbit is only increased 6", and the longitude of the node is diminished 4' 40", the 
other elements remaining the same, the errors distributed among all the latitudes 
will be reduced to a few seconds, and the longitudes will only be affected by the 
smallest errors, which will themselves be almost reduced to nothing, if, in addition. 
2" is taken from the epoch of the longitude. 



THIKD SECTION. 



THE DETERMINATION OF AN ORBIT SATISFYING AS NEARLY AS POSSIBLE ANY 
NUMBER OF OBSERVATIONS WHATEVER. 



172. 

If the astronomical observations and other quantities, on which the computa- 
tion of orbits is based, were absolutely correct, the elements also, whether deduced 
from three or four observations, would be strictly accurate (so far indeed as the 
motion is supposed to take place exactly according to the laws of Keplee), and, 
therefore, if other observations were used, they might be confirmed, but not cor- 
rected. But since all our measurements and observations are nothing more than 
approximations to the truth, the same must be true of all calculations resting 
upon them, and the highest aim of all computations made concerning concrete 
phenomena must be to approximate, as nearly as practicable, to the truth. But 
this can be accomplished in no other way than by a suitable combination of 
more observations than the number absolutely requisite for the determination of 
the unknown quantities. This problem can only be properly undertaken when 
an approximate knowledge of the orbit has been already attained, which is after- 
wards to be corrected so as to satisfy all the observations in the most accurate 
manner possible. 

It then can only be worth while to aim at the highest accuracy, when the 
final correction is to be given to the orbit to be determined. But as long as it 
appears probable that new observations will give rise to new corrections, it will 
be convenient to relax more or less, as the case may be, from extreme precision, 
if in this way the length of the computations can be considerably diminished. 
We win endeavor to meet both cases. 

32 (249) 



250 DETERMINATION OF AN ORBIT FROM [BoOK II. 

173. 

Ill the first place, it is of the greatest importance, that the several positions of 
the heavenly body on which it is proposed to base the orbit, should not be 
taken from single observations, but, if possible, from several so combined that the 
accidental errors might, as far as may be, mutually destroy each other. Obser- 
vations, for example, such as are distant from each other by an interval of a few 
days, — or by so much, in some cases, as an interval of fifteen or twenty days, — 
are not to be used in the calculation as so many different positions, but it would 
be better to derive from them a single place, wliich would be, as it were, a mean 
among all, admitting, therefore, much greater accuracy than single observations 
considered separately. This process is based on the following principles. 

The geocentric places of a heavenly body computed from approximate ele- 
ments ought to differ very little from the true places, and the differences between 
the former and latter should change very slowly, so that for an interval of a 
few days they can be regarded as nearly constant, or, at least, the changes may 
be regarded as proportional to the times. If, accordingly, the observations should 
be regarded as free from all error, the differences between the observed places 
corresponding to the times t, t', if', if", and those which have been computed from 
the elements, that is, the differences between the observed and the computed 
longitudes and latitudes, or right ascensions and declinations, would be quanti- 
ties either sensibly equal, or, at least, uniformly and very slowly increasing or de- 
creasing. Let, for example, the observed right ascensions a, a', a", a", etc., cor- 
respond to those times, and let a -\-d, a' -\- d', a" -\- d'\ a" -\- d'", etc., be the 
computed ones ; then the differences d, d', d", d'", etc. will differ from the true 
deviations of the •elements so far only as the observations themselves are errone- 
ous : if, therefore, these deviations can be regarded as constant for aU these ob- 
servations, the quantities d, d', d", d'", etc. will furnish as many different determi- 
nations of the same quantity, for the correct value of which it will be proper to 
take the arithmetical mean between those determinations, so far, of course, as 
there is no reason for preferring one to the other. But if it seems that the same 
degree of accuracy cannot be attributed to the several observations, let us assume 



Sect. 3.]' any number of observations. 251 

that the degree of acc-uracj in each may be considered proportional to the num- 
bers e, /, e", /", etc. respectively, that is, that errors reciprocally proportional to 
these numbers could have been made in the observations with equal facility; 
then, according to the principles to be propounded below, the most probable 
mean value will no longer be the simple arithmetical mean, but 

e e g -f ^e'8' -[- d'd'^' + e"'e"'b"' + etc. 
ee-\- e'e! -f- eV -|- e"'e"' -(-etc. 

Putting now this mean value equal to A, we can assume for the true right ascen- 
sions, a -[- (^ — A,a! -\- cf — J, a' \ 8" — ■ A, a!"\- 8'" — A, respectively, and then 
it will be arbitrary, which we use in the calculation. But if either the observa- 
tions are distant from each other by too great an interval of time, or if suffi- 
ciently approximate elements of the orbit are not yet known, so that it would 
not be admissible to regard their deviations as constant for all the observations, it 
will readily be perceived, that no other difference arises from this except that the 
mean deviation thus found cannot be regarded as common to all the observa- 
tions, but is to be referred to some intermediate time, which must be derived from 
the individual times in the same manner as A from the corresponding deviations, 
and therefore generally to the time 

eetA^ e'e't' - {- e"e"t" -A^ e"'e"'t"' -|- e tc. 
e e -}- e'e' -\- e"e" -\-e"'e"' -\-<iic. ' 

Consequently, if we desire the greatest accuracy, it will be necessary to compute 
the geocentric place from the elements for the same time, and afterwards to free 
it from the mean error A, in order that the most accurate position may be ob- 
tained. But it will in general be abundantly sufficient if the mean error is 
referred to the observation nearest to the mean time. What we have said -here 
of right ascensions, applies equally to declinations, or, if it is desired, to longitudes 
and latitudes : however, it will always be better to compare the right ascensions 
and declinations computed from the elements immediately with those observed ; 
for thus we not only gain a much more expeditious calculation, especially if we 
make use of the methods explained in articles 53-60, but this method has the 
additional advantage, that the incomplete observations can also be made use of; 
and besides, if every thing should be referred to longitudes and' latitudes, there 



252 DETERMINATION OF AN ORBIT FROM [BoOK II 

would be cause to fear lest an observation made correctly in right ascension, 
but badly in declination (or the opposite), should be vitiated in respect to both 
longitude and latitude, and thus become wholly useless. The degree of precision 
to be assigned to the mean found as above will be, according to the principles to 
be explained hereafter, 

Sl{ee-^ e'e' -{- e"e" -{- e"'e"' + etc.) ; 

so that four or nine equally exact observations are required, if the mean is to 
possess a double or triple accuracy. 

174. 

If the orbit of a heavenly body has been determined according to the methods 
given in the preceding sections from three or four geocentric positions, each one 
of which has been derived, according to the precepts of the preceding article, 
from a great many observations, that orbit will hold a mean, as it were, among 
all these observations ; and in the differences between the observed and computed 
places there will remain no trace of any law, which it would be possible to re- 
move or sensibly diminish by a correction of the elements. Now, when the whole 
number of observations does not embrace too great an interval of time, the best 
agreement of the elements with all the observations can be obtained, if only 
three or four normal positions are judiciously selected. How much advantage 
we shall derive from this method in determining the orbits of new planets or 
comets, the observations of which do not yet embrace a period of more than 
one year, will depend on the nature of the case. When, accordingly, the orbit 
to be determined is inclined at a considerable angle to the ecliptic, it will be 
m general based upon three observations, which we shall take as remote from 
each other as possible : but if in this way we should meet with any one of the 
cases excluded above (articles 160-162), or if the inclination of the orbit should 
seem too small, we shall prefer the determination from four positions, which, also, 
we shall take as remote as possible from each other. 

But when we have a longer series of observations, embracing several years, 
more normal positions can be derived from them ; on which account, we should 



Sect. 3.] any number of observations. 253 

not insure the greatest accuracy, if we were to select three or four positions only 
for the determination of the orbit, and neglect all the rest. But in such a case, 
if it is proposed to aim at the greatest precision, we shall take care to collect 
and employ the greatest possible number of accurate places. Then, of course, 
more data will exist than are required for the determination of the unknown 
quantities : but all these data will be liable to errors, however small, so that it 
will generally be impossible to satisfy all perfectly. Now as no reason exists, 
why, from among those data, we should consider any six as absolutely exact, but 
since we must assume, rather, upon the principles of probability, that greater or 
less errors are equally possible in all, promiscuously ; since, moreover, generally 
speaking, small errors oftener occur than large ones ; it is evident, that an orbit 
which, while it satisfies precisely the six data, deviates more or less from the 
others, must be regarded as less consistent with the principles of the calculus of 
probabilities, than one which, at the same time that it differs a little from those 
six data, presents so much the better an agreement with the rest. The investigar 
tion of an orbit having, strictly speaking, the maximum probability, will depend 
upon a knowledge of the law according to which the probability of errors de- 
creases as the errors increase in magnitude: but that depends upon so many 
vague and doubtful considerations — physiological included — which cannot be 
subjected to calculation, that it is scarcely, and indeed less than scarcely, possible 
to assign properly a law of this kind in any case of practical astronomy. Never- 
theless, an investigation of the connection between this law and the most prob- 
able orbit, which we will undertake in its utmost generality, is not to be regarded 
as by any means a barren speculation. 

175. 

To this end let us leave our special problem, and enter upon a very general 
discussion and one of the most fruitful in every application of the calculus to 
natural philosophy. Let V, V, V", etc. be functions of the unknown quantities 
j», q, r. s, etc., ^ the nmnber of those functions, v the number of the unknown 
quantities ; and let us suppose that the values of the functions found by direct 
observation are V = M, V = M', V" = M", etc. Generally speaking, the 



254 DETERmXATIOX OF AN ORBIT FROM [BoOK 11. 

determination of the unknown quantities will constitute a problem, indetermi- 
nate, determinate, or more than determinate, according as ii<iv, /^ = v, or 
/,< > v:'' We shall confine ourselves here to the last case, in which, evidently, an 
exact representation of all the observations would only be possible when they 
were all absolutely free from error. And since this cannot, in the nature of 
things, happen, every system of values of the unknown quantities p, q, r, s, etc., 
must be regarded as possible, which gives the values of the functions V — M, 
V — M', V" — M", etc., within the limits of the possible errors of observation ; 
this, however, is not to be understood to imply that each one of these systems 
would possess an equal degree of probability. 

Let us suppose, in the first place, the state of things in all the observations to 
have been such, that there is no reason why we should suspect one to be less 
exact than another, or that we are bound to regard errors of the same magnitude 
as equally probable in all. Accordingly, the probability to be assigned to each 
error J will be expressed by a function of J which we shall denote by ^> A. Now 
although we cannot precisely assign the form of this function, we can at least 
af&rm that its value should be a maximum for ^ = 0, equal, generally, for equal 
opposite values of A, and should vanish, if, for A is taken the greatest error, or a 
value greater than the greatest error: 9)//, therefore, would appropriately be re- 
ferred to the class of discontinuous functions, and if we undertake to sub.'^titute 
any analytical function in the place of it for practical purposes, this must be of 
such a form that it may converge to zero on both sides, asymptotically, as it were, 
from // = 0, so that beyond this limit it can be regarded as actually vanishing. 
Moreover, the probability that an error lies between the limits A and A -\- AA 
differing from each other by the infinitely small difference d z/, will be expressed 
byy^dz/; hence the probability generally, that the error lies between i> and 

* If, in the third case, the functions V, V, V" should be of such a nature that ju -j- 1 — 1' of them, 
or more, might be regarded as functions of the remainder, the problem would still be more than determi- 
nate with respect to these functions, but indeterminate with respect to the quantities p, q, r, s, etc. ; that 
is to say, it would be impossible to determine the values of the latter, even if the values of the func- 
tions V, V, V", etc. should be given with absolute exactness : but we shall exclude this case from our 
discussion. 



Sect. 3.] any numbee of obseryations. 255 

i)', will be given by the integral / 9) //.dz/ extended from J = D to J=^D'. 
This integral taken from the greatest negative value of z/ to the greatest positive 
value, or more generally from z/ = — co to z/ = -[-<» must necessarily be equal 
to unity. Supposing, therefore, any determinate system of the values of the 
quantities p, q, r, s, etc., the probability that observation would give for V the 
value M, will be expressed by 9 {M — V), substituting in V for p, q, r, s, etc., 
their values; in the same manner cp {M' — F), (f [M" — V"), etc. will express the 
probabilities that observation would give the values M', M", etc. of the func- 
tions V, V", etc. Wherefore, since we are authorized to regard all the observa- 
tions as events independent of each other, the product 

^ ^M—V) (f [M'—V) if {M"—V") etc., =i2 
will express the expectation or probability that all those values will result to- 
gether from observation. - 

176. 

Now in the same manner as, when any determinate values whatever of the 
unknown quantities being taken, a determinate probability corresponds, previ- 
ous to observation, to any system of values of the functions F, V, V", etc. ; so, 
inversely, after determinate values of the functions have resulted from observa- 
tion, a determinate probability will belong to every system of values of the un- 
known quantities, from which the values of the functions could possibly have 
resulted : for, evidently, those systems will be regarded as the more probable in 
which the greater expectation had existed of the event which actually occurred. 
The estimation of this probability rests upon the following theorem : — 

If, any hjpoiliesis H leing made, the probability of any determinate event E is h, and 
if, another hypothesis H' being made excluding the former and equally probahle in itself, the 
probability of the same event is h' ; then I say, tvhen the event E has actually occurred, thai 
the probability that H tvas the true hypothesis, is to the probability that H' was the true 
hypothesis, as h to \x. 

For demonstrating which let us suppose that, by a classification of all the cir- 
cumstances on which it depends whether, with H or H' or some other hypothesis, 



256 



DETERMINATION OF AN ORBIT FROM 



[Book II. 



the event E or some other event; should occur, a system of the different cases is 
formed, each one of which cases is to be considered as equally probable in itself 
(that is, as long as it is uncertain whether the event E, or some other, will occur), 
and that these cases be so distributed. 



that among them 
may be found 


in which should be assumed 
the hypothesis 


in such a mode as would give 
occasion to the event. 


m 


H 


E 


n 


H 


different from E 


m' 


H' 


E 


n' 


H' 


different from E 


rri' 


different from H and H' 


E 


7i' 


different from ^and H' 


different from E 



Then we shall have 



h = 



',' m -\-n ' 

moreover, before the event was known the probability of the hypothesis H was 

m-\-n 
m -^ n -\- n^ -\- n' -\- rd' -\- ri' ' 

but after the event is known, when the cases n, n', if disappear from the number 
of the possible cases, the probability of the same hypothesis will be 



7n-\-m -\-m ' 

in the same way the probability of the hypothesis H' before and after the event, 
respectively, will be expressed by 



.'JrW 



and 



m -\- n -\- m! -\- n' -\- ??/' -|- Ji" " " m-\- m' -\- 
since, therefore, the same probability is assumed for the hypotheses R and H' 
before the event is known, we shall have 

m -\-n = m^ -\- n', 

whence the truth of the theorem is readily inferred. 

Now, so far as we suppose that no other data exist for the determination of 
the unknown quantities besides the observations V=M, V' = M', V":^M", 



Sect. 3.] a^y number of observations. 257 

etc., and, therefore, that all systems of values of these unknown quantities were 
equally probable previous to the observations, the probability, evidently, of any 
determinate system subsequent to the observations will be proportional to S2. 
This is to be understood to mean that the probability that the values of the un- 
known quantities lie between the infinitely near limits j» and jo-j-djo, q and g-\-dq, 
r and r-j- dr, s and 5 -|- ^^j ^tc. respectively, is expressed by 

Xi2dj!?d^drd5 , etc., 

where the quantity 1 will be a constant quantity independent of p, q, r, s, etc. : 
and, indeed, ^ will, evidently, be the value of the integral of the order v, 

f^£ldipdiqdrds , etc., 

for each of the variables p, q, r, 5, etc., extended from the value — 00 to the 
value -[- GO • 

177. 

Now it readily follows from this, that the most probable system of values of 
the quantities p, q, r, s, etc. is that in which il acquu-es the maximum value, and, 
therefore, is to be derived from the v equations 

:^ = 0, -— = 0, -5—= 0, -7- = 0, etc. 
These equations, by putting 

V—M=:v, r—M' = v', V"—M" = v", etc., and ^^, = ^>' ^, 
assume the following form : — 

d^9^ + d^9^+a^9^ +etc.= 0, 
r^^'^ + d^^'^+d^?'^ +etc.= 0, 

Av f . Av' , , . Avl' , ,f . . „ 

■^r'f ^^J-rf^ +d79^^ -l-etc.= 0, 
i ■ 

Av , . Av' , , , Av" r rr \ , a 

dl9'^ + d79'^+d79^^ +etc.= 0. 
Hence, accordingly, a completely determinate solution of the problem can be 
obtained by elimination, as soon as the nature of the function 9' is known. Since 

33 



258 DETERJMINATION OF AN ORBIT FROM [BoOK 11. 

this cannot be defined a priori, we will, approaching the subject from another 
point of view, inqnh-e upon what function, tacitly, as it were, assumed as a 
base, the common principle, the excellence of which is generally acknowledged, 
depends. It has been customary certainly to regard as an axiom the hypothesis 
that if any quantity has been determined by several direct observations, made 
under the same circumstances and with equal care, the arithmetical mean of the 
observed values affords the most probable value, if not rigorously, yet very 
nearly at least, so that it is always most safe to adhere to it. By putting, 
therefore, 

F=F'=F"etc.=j9, 
we ought to have in general, 

9' {M—p) -f 9' {M' —p) -f ^>' [M" — p)-\- etc. = 0, 
if instead of p is substituted the value 

-(ilf-f ilf' + Jf"+etc.), 
wnatever positive integer }i expresses. By supposing, therefore, 

M' = M''= etc. =M—iiN, 
we shall have in general, that is, for any positive integral value of \i, 

whence it is readily inferred that ^ must be a constant quantity, which we will 
denote by k. Hence we have 

log (^ A^=\'kA A -\- Constant, 

denoting the base of the hyperbolic logarithms by e and assuming 

Constant = log x. 
Moreover, it is readily perceived that I" must be negative, in order that S2 may 
really become a maximum, for which reason we shall put 

and since, by the elegant theorem first discovered by Laplace, the integral 



Sect. 3.] any number of observations. 259 

from J :^ — oo to// = -|-oo is^, (denoting bj n the semicircumference of 
the circle the radius of which is unity), our function becomes 

178. 

The function just found cannot, it is true, express rigorously the probabilities 
of the errors: for since the possible errors are in all cases confined within certain 
limits, the probability of errors exceeding those limits ought always to be zero, 
while our formula always gives some value. However, this defect, which every 
analytical function must, from its nature, labor under, is of no importance in 
practice, because the value of our function decreases so rapidly, when hJ has 
acquired a considerable magnitude, that it can safely be considered as vanishing. 
Besides, the nature of the subject never admits of assigning with absolute rigor 
the limits of error. 

Finally, the constant h can be considered as the measure of precision of the 
observations. For if the probability of the error J is supposed to be expressed 
in any one system of observations by 

_^ .-hhAA 

and in another system of observations more or less exact by 

the expectation, that the error of any observation in the former system is con- 
tained between the limits — d and -|- d will be expressed by the integral 



/ 



^^-hhA^^J 



taken from J = — dto J z=-^ d ; and in the same manner the expectation, that 
the error of any observation in the latter system does not exceed the limits — d' 
and -|- d' will be expressed by the integral 



I 



^g-h-h'AA^JJ 



extended from J = — d' to J = -\-d' : but both integrals manifestly become 



260 DETERMDsATION OF AN ORBIT FROM [BoOK II. 

equal when we have /^ cT = h'd'. Now, therefore, if for example // = 2 A, a double 
error can be committed in the former system with the same facility as a single 
error in the latter, in which case, according to the common way of speaking, a 
doable degree of precision is attributed to the latter observations. 

179. 

We will now develop the conclusions which follow from this law. It is evi- 
dent, in order that the product 

may become a maximum, that the sum 

vv -\- v'v' -\- v"v" -j- etc., 

must become a minimum. Therefore, tJiat will he the most prolable syste^n of values of 
the unknown quantities p, q, r, s, etc., in which the sum of the squares of the diff^et-emes 
between the observed and computed values of the functions V, V, Y", etc. is a minimum, if 
the same degree of accuracy is to be presumed in all the observations. This prin- 
ciple, which promises to be of most frequent use in all applications of the mathe- 
matics to natural philosophy, must, everywhere, be considered an axiom with 
the same propriety as the arithmetical mean of several observed values of the 
same quantity is adopted as the most probable value. 

This principle can be extended without difficulty to observations of unequal 
accuracy. K, for example, the measures of precision of the observations by 
means of which V=^M, V = M', V" =iM", etc. have been found, are expressed, 
respectively, by h, h', h", etc., that is, if it is assumed that errors reciprocally pro- 
portional to these quantities might have been made with equal facility in those 
observations, this, evidently, will be the same as if, by means of observations of 
equal precision (the measure of which is equal to unity), the values of the func- 
tions hV, h'V, h"V", etc., had been directly found to be hM, // Jf' , A"ilf ", etc. : 
wherefore, the most probable system of values of the quantities 2h §"? '"? •^j ^tc, 
will be that in which the sum of hhvv -|- liliv'v' -\- li'H'v'v" -\- etc , that is, in tvhich 
the sum of the squares of the differences between the actuall// observed and computed values 
)nuUipllcd by numbers that measure the degree of precision, is a minimum. In this way it 



Sect. 3.] any nibiber of observations. 261 

is not even necessary that the functions V, V, V", etc. relate to homogeneous 
quantities, but thej may represent heterogeneous quantities also, (for example, 
seconds of arc and time), provided only that the ratio of the errors, which might 
have been committed with equal faciHty in each, can be estimated. 



180. 

The principle explained in the preceding article derives value also from this, 
that the numerical determination of the unknown quantities is reduced to a very 
expeditious algorithm, when the functions V, V, V'\ etc. are linear. Let us 
suppose 

V — Jtf = v=^ — m -\- a2} -\- h q -\- cr -\- ds -\- etc. 
V — M' = y' = — m' -\- a'p -\- h'q -j- cr -j- d's -\- etc. 
V" — M"^=v"=^ — m"-\- a"p -|- h"q -j- c"r -\- d"s -\- etc. 
etc., and let us put 

av -\- a'v' -\- a"v" -\- etc. = P 
hv -\- b'v' -f b"v" + etc. = Q 
ev -\- c'v' -j- c"v" -|- etc. ^ R 
dv + d'v' + d"v" + etc. = >S' 
etc. Then the v equations of article 177, from which the values of the unknown 
quantities must be determined, will, evidently, be the following : — 

P = 0, Q=^,R=^%S=^, etc., 
provided we suppose the observations equally good ; to which case we have shown 
in the preceding article how to reduce the others. We have, therefore, as many 
linear equations as there are unknown quantities to be determined, from which 
the values of the latter will be obtained by common elimination. 

Let us see now, whether this elimination is always possible, or whether the 
solution can become indeterminate, or even impossible. It is known, from the 
theory of elimination, that the second or third case will occur when one of the 
equations 

P=% Q = 0,B=0, S=0, etc., 
being omitted, an equation can be formed from the rest, either identical with the 



262 DETERMINATION OF AN OllBIT FKOM [BoOK II. 

omitted one or inconsistent with it, or, which amounts to the same thing, when 
it is possible to assign a linear function 

aP^i^Q^yR-y dS + etc., 
which is identically either equal to zero, or, at least, free from all the unknown 
quantities /», q, r, s, etc. Let us assume, therefore, 

aP-\-(SQ-]-yB-\-dS-{- etc. = ;c. 
We at once have the identical equation 

[v -\-m)v-\- {v' -|- m) v' -\-{v" -\- m") v" -\- etc. = fP -\'<lQ~\- ^-^ H~ -^'^ H~ ^tc. 
If, accordingly, by the substitutions 

p = ax, q = i^x, r = yx, s = dx, etc. 
we suppose the functions v, v', v", to become respectively, 

— m-\-lx, — m -\- I'x, — m" -{- H'x, etc., 
we shall evidently have the identical equation 

(^ ^ -(- Vl' -J- I" I" -\- etc.) XX — {Ifn-^ I'm' -|- Vnf etc.) x=^y.x, 
that is, 

X ;, _^ r).' -I- I'T + etc. = 0, >t 4- ?. m -f- I'm' + l"m" + etc. = : 
hence it must follow that X ^ 0, ^' = 0, "k" = 0, etc. and also x = 0. Then it is 
evident, that all the functions V, V V", are such that their values are not 
changed, even if the quantities p, q, r, s, etc. receive any increments or decre- 
ments whatever, proportional to the numbers a, fi, y, d, etc. : but we have already 
mentioned before, that cases of this kind, in which evidently the determination 
of the unknown quantities would not be possible, even if the true values of the 
functions V, V, V", etc., should be given, do not belong to this subject. 

Finally, we can easily reduce to the case here considered, all the others in 
which the functions V, V, V", etc. are not linear. Letting, for instance, tt, x, {>, 
o, etc., denote approximate values of the unknown quantities jk», §-, r, s, etc., (which 
we shall easily obtain if at first we only use v of the fi equations V=3f, V'^=M', 
V" z= M", etc.), we will introduce in place of the unknown quantities the others, 
p', q, r, /, etc., putting p=zn -\-p>', q-= x -[- 4i r = () -f- /, s = a -|- 5', etc. : the 
values of these new unknown quantities will evidently be so small that their 



Sect. 3.] any number of observations. 263 

squares and products may be neglected, by which means the equations become 
Hnear. If, after the calculation is completed, the values of the unknown quanti- 
ties p, q, r', s', etc., prove, contrary to expectation, to be so great, as to make it 
appear unsafe to neglect the squares and products, a repetition of the same pro- 
cess (the corrected values of p, q, r, s, etc. being taken instead of n, /_, q, o, etc.), 
will furnish an easy remedy. 

181. 

When we have only one unknown quantity p, for the determination of which 
the values of the functions ap -\- n, ap -\- n, d'p -\- n\ etc. have been found, re- 
spectively, equal to M, M', M", etc., and that, also, by means of observations 
equally exact, the most probable value of p will be 

. am -j- a'm' -|- aW -|- etc. 

aa-\- a! a' -j- aV -\- etc. ' 

putting m, m', m", respectively, for M — n, M' — n', M" — n", etc. 

In order to estimate the degree of accuracy to be attributed to this value, let 
us suppose that the probability of an error z/ in the observations is expressed by 

\Jn 
Hence the probability that the true value of p is equal to A -\-p will be propor- 
tional to the function 

g-AA ({ap—mf-\-{a'p~m'f+{a"p-m"f+ etc.) 

if A -\-p is substituted for p. The exponent of this function can be reduced to 
the form, 

— hh {aa -\- dd -f d'd' -f etc.) ^pp —.2pA-}- B), 
in which B is independent of p : therefore the function itself will be propor- 
tional to 

- — hh(aa-\-a'a'-{-a"a"-\-etD.) p'p' 

It is evident, accordingly, that the same degree of accuracy is to be assigned to 
the value A as if it had been found by a direct observation, the accuracy of which 
would be to the accuracy of the original observations a.s h^{aa-\-dd'^d'd'-\- etc.) 
to h, or as ^ (aa -f- dd -j- dW -{- etc.) to unity. 



264 DETERMNATION OF AN OllBIT FEOM [BoOK II. 

182. 

It will be necessary to preface the discussion concerning the degree of accu- 
racy to be assigned to the values of the unknown quantities, when there are sev- 
eral, with a more careful consideration of the function vv -{- vV -{- v"v" -{- etc., 
which we will denote by W. 

I. Let us put 

also 

a 

and it is evident that we have p' = P, and, since 

&W__AW '2p'Ap' ^ 

Ap djo a d/7 ' 

that the function W is independent of jo. The coefficient a = aa -\- a a -\- a" a" -\- 
etc. will evidently always be a positive quantity. 
n. In the same manner we will put 



also 



and we shall have 



* ^-^ / = ^' + ^^V + 7'r + d's + etc., 



W'—^^=W", 



^'=i^_ii^=:^_^-/, and'-^'=0, 

whence it is evident that the function W" is independent both of 7; and q. 
This would not be so if ^' could become equal to zero. But it is evident 
that W is derived from vv -\- v'v -\-v"v" -\- etc., the quantity p being eliminated 
from V, v, v", etc., by means of the equation j»' = ; hence, {^ will be the sum of 
the coefficients oi qq in vv, v'v', v"v", etc., after the elimination; each of these 
coefficients, in feet, is a square, nor can all vanish at once, except in the case 
excluded above, in which the unknown quantities remain indeterminate. Thus 
it is evident that §' must be a positive quantity. ♦- 



Sect. 3.] any number of observations. 265 

III. By putting again, 

^^ = / = r + /V + rs + etc., and W' — y= W'% 



we shall have 






also W" independent of p, and g, as well as r. Finally, that the coeflficient of /" 
must be positive is proved in the same manner as in II. In fact, it is readily per- 
ceived, that y" is the sum of the coefficients oi rr mvv, vv', v"v", etc., after the 
quantities p and q have been eliminated from v, v', v", etc., by means of the equa- 
tions p' = 0, / = 0. 

IV. In the same way, by putting 



ds 
we shall have 



J^' = / = r' + r'5 + etc., W^=W"'—%, 



= S--p-j,q-^r, 



W'-'^ independent of p, q, r, s, and d'" a positive quantity. 

V. In this manner, if besides p, q, r, s, there are still other unknown quanti- 
ties, we can proceed further, so that at length we may have 

W= — p'p -\- -XT 44 "h —r ^'^'' -|- Y" ^'^' ~^ ^^^' ~i~ Constant, 

in which all the coefficients will be positive quantities. 

VI. Now the probability of any system of determinate values for the quan- 
tities p, q, r, s, etc. is proportional to the function e~*^^; wherefore, the value of 
the quantity p remaining indeterminate, the probability of a system of determi- 
nate values for the rest, will be proportional to the integral 

/e-^''^djo 

extended from jt?=:r — oo to jk»=-|-°° ? which, by the theorem of Laplace, becomes 

therefore, this probability will be proportional to the function e~'''''-^\ In the 
same manner, if, in addition, q is treated as indeterminate, the probability of a 

34 



260 DETERMINATION OF AN ORBIT FROM [BoOK 11. 

system of determinate values for r, s, etc. will be proportional to the integral 

extended from q = — co upto§'=-|-oo, which is 

or proportional to the function e"*'''*^". Precisely in the same waj^, if r also is 
considered as indeterminate, the probability of the determinate values for the rest, 
s, etc. will be proportional to the function e~''''^^"', and so on. Let us suppose the 
number of the unknown quantities to amount to four, for the same conclusion 
will hold good, whether it is greater or less. The most probable value of s will 
be — yr,, and the probability that this will differ from the truth by the quantity 
a, will be proportional to the function e~^'''^''"'; whence we conclude that the 
measure of the relative precision to be attributed to that determination is ex- 
pressed by ^d'", provided the measure of precision to be assigned to the original 
observations is put equal to unity. 

183. 

By the method of the preceding article the measure of precision is conven- 
iently expressed for that unknown quantity only, to which the last place has 
been assigned in the work of elimination ; in order to avoid which disadvantage, 
it will be desirable to express the coefl&cient d'" in another manner. From the 
equations 

p=/ 

it follows, that p', q', r, s', can be thus expressed by means of P, Q, R, 8, 
p' = P 



SeCJ, 3.] ANY NUMBER OF OBSERVATIONS. 267 

so that % 5(', 23', W, 23", ^" may be determinate quantities. We shall have, 
therefore (by restricting the number of unknown quantities to four), 

r' r p I 33"^ , s" pile. 

Hence we deduce the following conclusion. The most probable values of the 

unknown quantities p, q, r, s, etc., to be derived by elimination from the equations 

P=z{), Q = Q, R=0, S=0, etc., 

will, if P, Q, R, S, etc., are regarded for the time as indeterminate, be expressed 
in a linear form by the same process of elimination by means of P, Q, R, S, etc., 
so that we may have 

p = L-\-AP^BQ^CR-^DS-\- etc. 

q=.L'-^AP-^B'Q^C'R^D'-S-^ etc. 

r=L"^A'P^B"Q^G"R^D"S^ etc. 

s=L"'-]-^"P-^B"'Q^-0"'R^D"'S-\- etc. 

etc. 
This being done, the most probable values of p, q, r, s, etc., will evidently be 
L, U, L", 11" , etc., respectively, and the measure of precision to be assigned to 
these determinations respectively will be expressed by 

J_ J_ J_ _2_ 

V3' y/^' v/<^'" V/^"" ' 

the precision of the original observations being put equal to unity. That which 
we have before demonstrated concerning the determination of the unknown 
quantity s (for which -ryj-, answers to D'"') can be applied to all the others by the 
simple interchange of the unknown quantities. 

184. 

In order to illustrate the preceding investigations by an example, let us sup- 
pose that, by means of observations in which equal accuracy may be assumed, 
we have found 



268 DETERmNATION OF AN ORBIT FROM [BoOK II. 

3jo + 2^ — 5r=5 
4|.+ ^+4r=21, 

but from a fourth observation, to which is to be assigned one half the same 
accuracy only, there results 

— 2/>+6^ + 6r = 28. 

"We will substitute in place of the last equation the following : — 

— JO -|- 3 ^ + 3 r = 14, 

and we will suppose this to have resulted from an observation possessing equal 
accuracy with the former. Hence we have 

P = 27jo+ 6^ — 88 

^= 6j(?+15^ + r — 70 

and hence by elimination, 

19899j(? = 49154 + 809P — 324 Q^^R 
131 q= 2617— 12 P+ 54§ — P 
6633 r = 12707 4- 2P— 9 ^ + 123 i?. 
The most probable values of the unknown quantities, therefore, will be 

j» = 2.470 
^ = 3.551 
r = 1.916 

and the relative precision to be assigned to these determinations, the precision of 
the original observations being put equal to unity, will be 

„ / 19899 . ^^ 

for ^ \/ ^ = 3.69 

forr \/^ = 7.34. 



Sect. 3.] J^y number of observations. 



185. 

The subject we have just treated might give rise to several elegant analytical 
investigations, upon which, however, we will not dwell, that we may not be too 
much diverted from our object. For the same reason we must reserve for another 
occasion the explanation of the devices by means of which the numerical calcu- 
lation can be rendered more expeditious. I will add only a single remark. 
When the number of the proposed functions or equations is considerable, the 
computation becomes a little more troublesome, on this account chiefly, that the 
coefficients, by which the original equations are to be multiplied in order to ob- 
tain P, Q, a, jS, etc., often involve inconvenient decimal fractions. If in such 
a case it does not seem worth while to perform these multiplications in the most 
accurate manner by means of logarithmic tables, it will generally be sufficient 
to employ in place of these multipliers others more convenient for calculation, 
and differing but little from them. This change can produce sensible errors in 
that case only in which the measure of precision in the determination of the 
unknown quantities proves to be much less than the precision of the original 
observations. 

186. 

In conclusion, the principle that the sum of the squares of the differenc'CS 
between the observed and computed quantities must be a minimum may, in the 
following manner, be considered independently of the calculus of probabilities. 

When the number of unknown quantities is equal to the number of the ob- 
served quantities depending on them, the former may be so determined as exactly 
to satisfy the latter. But when the number of the former is less than that of the 
latter, an absolutely exact agreement cannot be obtained, unless the observations 
possess absolute accuracy. In this case care must be taken to estabhsh the best 
possible agreement, or to diminish as far as practicable the differences. This idea, 
however, from its nature, involves something vague. For, although a system of 
values for the unknown quantities which makes all the differences respectively 



270 DETERMINATION OF AN ORBIT FROM [BOUK II. 

less than another sj^stem, is without doubt to be preferred to the latter, still the 
choice between two systems, one of which presents a better agreement in some 
observations, the other in others, is left in a measure to our judgment, and innu- 
merable different principles can be proposed by which the former condition is 
satisfied. Denoting the differences between observation and calculation by J, 
J', J", etc., the first condition will be satisfied not only if //z/ -j- /I' J' -{- J" J" -\- 
etc, is a minimum (which is our principle), but also if J^-\-J'*-\- J"* -\- etc., or 
J^ -\- J"^ -\- J"^ -\- etc., or in general, if the sum of any of the powers with an 
even exponent becomes a minimuiu. But of all these principles ours is the most sim- 
ple ; by the others we should be led into the most complicated calculations. 

Our principle, which we have made use of since the year 1795, has lately 
been published by Legendke in the work Noiivelles methodes pour la determination des 
orbit es des cometes, Paris, 1806, where several other properties of this principle have 
been explained, which, for the sake of brevity, we here omit. 

If we were to adopt a power with an infinite even exponent, we should be 
led to that system in which the greatest dijEferences become less than in any other 
system. 

Laplace made use of another principle for the solution of linear equations the 
number of which is greater than the number of the unknown quantities, which 
had been previou.sly proposed by Boscovich, namely, that the sum of the errors 
themselves taken positively, be made a minimum. It can be easily shown, that a 
system of values of unknown quantities, derived from this principle alone, must 
necessarily* exactly satisfy as many equations out of the number proposed, as 
there are unknown quantities, so that the remaining equations come under consid- 
eration only so far as the}- help to determine the choice : if, therefore, the equation 
V ^=. M, for example, is of the number of those wdiich are not satisfied, the sys- 
tem of values found according to this principle would in no respect be changed, 
even if any other value N had been observed instead of 31, provided that, denot- 
ing the computed value by n, the difierences M — n, N — n, were affected by the 
same signs. Besides, Laplace qualifies in some measure this principle by adding 

* Except the special cases in which the problem remains, to some extent, indeterminate. 



Sect. 3.] any number of observations. 271 

a new condition : he requires, namely, that the sum of the differences, the signs 
remaining unchanged, be equal to zero. Hence it follows, that the number of 
equations exactly represented may be less by unity than the number of unknown 
quantities ; but what we have before said will still hold good if there are only 
two unknown quantities. 

187. 

From these general discussions we return to our special subject for the sake 
of which they were undertaken. Before the most accurate determination of 
the orbit from more observations than are absolutely requisite can be com- 
menced, there should be an approximate determination which will nearly satisfy 
all the given observations. The corrections to be applied to these approximate 
elements, in order to obtain the most exact agreement, will be regarded as the 
objects of the problem. And when it can be assumed that these are so small 
that their squares and products may be neglected, the corresponding changes, 
produced in the computed geocentric places of a heavenly body, can be obtained 
by means of the differential formulas given in the Second Section of the First 
Book. The computed places, therefore, which we obtain from the corrected ele- 
ments, will be expressed by linear functions of the corrections of the elements, 
and their comparison with the observed places according to the principles before 
explained, will lead to the determination of the most probable values. These 
processes are so simple that they require no further illustration, and it appears at 
once that any number of observations, however remote from each other, can 
be employed. The same method may also be used in the correction of the parcb- 
holic orbits of comets, should we have a long series of observations and the best 
agreement be required. 

188. 

The preceding method is adapted principally to those cases in which the 
greatest accuracy is desired: but cases very frequently occur where we may, 
without hesitation, depart from it a little, provided that by so doing the calcula- 



272 



DETERMINATION OF AX ORBIT FROM 



[Book II. 



tion is considerably abridged, especially when the observations do not embrace a 
groat interval of time ; here the final determination of the orbit is not yet 
proposed. In such cases the following method may be employed with great 
a'^viuitage. 

Let complete places L and L' be selected from the whole nmnber of observa- 
tions, and let the distances of the heavenly body from the earth be computed 
from the approximate elements for the corresponding times. Let three hypothe- 
ses then be framed with respect to these distances, the computed values being 
retained in the first, the first distance being changed in the second hypothesis, 
and the second in the third hypothesis ; these changes can be made in proportion 
to the uncertainty presumed to remain in the distances. According to these 
three hypotheses, which we present in the following table, 





Hyp. I. 


Hyp. II. 


Uyp. ni. 


Distance * corresponding to the first place, 
Distance corresponding to the second place, 


D 
D' 


D' 


D 

iX + S 



let three sets of elements be computed from the two places X, L', by the methods 
explained in the first book, and afterwards from each one of these sets the geo- 
centric places of the heavenly body corresponding to the times of all the remain- 
ing observations. Let these be (the several longitudes and latitudes, or right 
ascensions and declinations, being denoted separately), 

in the first set .... ilf. M', M", etc. 

in the second set . . . M-\- a, M' -\- a', M"-]- a", etc; 

in the third set . . . . M-\- /?, M' -\- (V, M"-\- fi", etc. 
Let, moreover, the observed 

places be respectively N, N', N", etc. 

Now, so far as proportional variations of the individual elements correspond 



* ?t will be still more convenient to use, instead of tlie distances themselves, the logarithms of the 
curtate distances. 



Sect. 3.] any number of observations. 273 

to small variations of tlie distances D, D', as well as of the geocentric places 
computed from tliem, we can assume, that the geocentric places computed from 
the fourth system of elements, based on the distances from the earth D -\-xd, 
D' -^y^', are respectively M^ax^^y, M -\-a'x-^ (i'y, M" -\- a"x -)- fy, etc. 
Hence, x, y, will be determined, according to the preceding discussions, in such a 
manner (the relative accuracy of the observations being taken into account), that 
these quantities may as far as possible agree with N, N', N'\ etc., respectively. 
The corrected system of elements can be derived either from L, I! and the dis- 
tances D -^x^, D' -\-x8', or, according to well-known rules, from the three first 
systems of elements by simple interpolation. 

189. 

This method differs from the preceding in this respect only, that it satisfies 
two geocentric places exactly, and then the remaining places as nearly as possi- 
ble ; while according to the other method no one observation has the preference 
over the rest, but the errors, as far as it can be done, are distributed among all. 
The method of the preceding article, therefore, is only not to be preferred to the 
former when, allowing some part of the errors to the places L, L', it is possible to 
diminish considerably the errors in the remaining places : but yet it is generally 
easy, by a suitable choice of the observations L, L', to provide that this difference 
cannot become very important. It will be necessary, of course, to take care that 
such observations are selected for L, L', as not only possess the greatest accuracy, 
but also such that the elements derived from them and the distances are not 
too much affected by small variations in the geocentric places. It will not, there- 
fore, be judicious to select observations distant from each other by a small inter- 
val of time, or those to which correspond nearly opposite or coincident heliocen- 
tric places. 

35 



FOURTH SECTION. 



ON THE DETERSIINATION OF ORBITS, TAKING INTO ACCOUNT THE 
PERTURBATIONS. 



190. 

The perturbations which the motions of planets suffer from the influence of . 
other planets, are so small and so slow that they only become sensible after a 
long interval of time ; within a shorter time, or even within one or several entire 
revolutions, according to circumstances, the motion would differ so little from the 
motion exactly described, according to the laws of Kepler, in a perfect ellipse, 
that observations cannot show the difference. As long as this is true, it would 
not be worth while to undertake prematurely the computation of the perturba- 
tions, but it will be sufficient to adapt to the observations what we may call an 
osculating conic section: but, afterwards, when the planet has been accurately 
observed for a longer time, the effect of the perturbations will show itself in such 
a manner, that it will no longer be possible to satisfy exactly all the observations 
by a purely elliptic motion ; then, accordingly, a complete and permanent agree- 
ment cannot be obtained, unless the perturbations are properly connected with 
the elll])tic motion. 

Since the determination of the elliptic elements with which, in order that the 
observations may be exactly represented, the perturbations are to be combined, 
supposes a knowledge of the latter ; so, inversely, the theory of the perturlDations 
cannot be accurately settled unless the elements are already very nearlj^ kno-\va : 
the nature of the case does not admit of this difficult task being accomplished 
with complete success at the first trial : but the perturbations and the elements 
can be brought to the highest degree of perfection only by alternate corrections 
(274) 



Sect. 4.] ON the determination of orbits. 275 

often repeated. Accordingly, the first theory of perturbations will be constructed 
upon those purely elliptical elements which have been approximately adjusted to 
the observations ; a new orbit will afterwards be investigated, which, with the 
addition of these perturbations, may satisfy, as far as practicable, the observa- 
tions. If this orbit differs considerably from the former, a second determination 
of the perturbations will be based upon it, and the corrections will be repeated 
alternately, until observations, elements, and perturbations agree as nearly as 
possible. 

191. 

Since the development of the theory of perturbations from given elements is 
foreign to our purpose, we will only point out here how an approximate orbit 
can be so corrected, that, joined with given perturbations, it may satisfy, in 
the best manner, the observations. This is accomplished in the most simple 
way by a method analogous to those which we have explained in articles 124, 
165, 188. The numerical values of the perturbations will be computed from the 
equations, for the longitudes in orbit, for the radii vectores, and also for the helio- 
centric latitudes, for the times of all the observations which it is proposed to use, 
and which can either be three, or four, or more, according to circumstances : for 
this calculation the materials will be taken from the approximate elliptic ele- 
ments upon which the theory of perturbations has been constructed. Then two 
will be selected from all the observations, for which the distances from the earth 
will be computed from the same approximate elements : these will constitute the 
first hypothesis, the second and third will be formed by changing these distances 
a little. After this, in each of the hypotheses, the heliocentric places and the 
distances from the sun will be determined from two geocentric places; from those, 
after the latitudes have been freed from the perturbations, will be deduced the 
longitude of the ascending node, the inclination of the orbit, and the longi- 
tudes in orbit. The method of article 110 with some modification is useful in 
this calculation, if it is thought worth while to take account of the secular varia- 
tion of the longitude of the node and of the inclination. If (i, /5', denote the 
heliocentric hi litudes freed from the periodical perturbations; ^, X', the heliocen- 



276 ON THE DETERMINATION OF ORBITS, [BoOK II. 

trie longitudes; Q, Q-\-J, the longitudes of the ascending node; i,i-{-d, the 
inclinations of the orbit ; the equations can be conveniently given in the follow- 
ing form : — 

tan (3 = tan i sin (X — Q), 

— f^] ^. tan (y = tan ^ sin (I' — J — 9,). 

tan {i-\-8) ' ^ ' 

This value of - — t^tt^t acquires all the requisite accuracy by substituting an 
approximate value for ^': i and Q> can afterwards be deduced by the common 
methods. 

Moreover, the sum of the perturbations will be subtracted from the longitudes 
in orbit, and also from the two radii vectores, in order to produce purely elliptical 
values. But here also the effect, which the secular variations of the place of the 
perihelion and of the eccentricity exert upon the longitude in orbit and radius 
vector, and which is to be determined by the differential formulas of Section I. 
of the First Book, is to be combined directly with the periodical perturbations, 
provided the observations are sufficiently distant from each other to make it 
appear worth while to take account of it. The remaining elements will be deter- 
mined from these longitudes in orbit and corrected radii vectores together with 
the corresponding times. Finally, from these elements will be computed the 
geocentric places for all the other observations. These being compared with the 
observed places, in the manner we have explained in article 188, that set of 
distances will be deduced, from which will follow the elements satisfying in the 
best possible manner all the remaining observations. 

192. 

The method explained in the preceding article has been prmcipally adapted 
to the determination of the first orbit, including the perturbations : but as soon 
as the mean elliptic elements, and the equations of the perturbations have both 
become very nearly known, the most accurate determination will be very con- 
veniently made with the aid of as many observations as possible by the method 
of article 187, which will not require particular explanation in this place. Now 
if the number of the best observations is sufficiently great, and a great interval 



Sect. 4.] taking into account the perturbations. 277 

of time is embraced, this method can also be made to answer in several cases for 
the more precise determination of the masses of the disturbing planets, at least 
of the larger planets. Indeed, if the mass of any disturbing planet assumed in 
the calculation of the perturbations does not seem sufficiently determined, besides 
the six unknown quantities depending on the corrections of the elements, yet 
another, fj,, will be introduced, putting the ratio of the correct mass to the assumed 
one SiS 1 -\- fi to 1 ; it will then be admissible to suppose the perturbations them- 
selves to be changed in the same ratio, whence, evidently, in each one of the com- 
puted places a new linear term, containing /a, will be produced, the development 
of which will be subject to no dif&culty. The comparison of the computed places 
with the observed according to the principles above explained, will furnish, at the 
same time with the corrections of the elements, also the correction ^. The 
masses of several planets even, which exert very considerable perturbations, can 
be more exactly determined in this manner. There is no doubt but that the mo- 
tions of the new planets, especially Pallas and Juno, which suffer such great per- 
turbations from Jupiter, may furnish in this manner after some decades of years, 
a most accurate determination of t:he mass of Jupiter ; it may even be possible 
perhaps, hereafter, to ascertain, from the perturbations which it exerts upon the 
others, the mass of some one of these new planets. 



APPENDIX. 



1.* 

The value of t adopted in the Solar Tables of Hansen and Olufsen, (Copen- 
hagen, 1853,) is 365.2563582. Using this and the value of \i, 

1 

^ 354936' 

from the last edition of Laplace's Systhne du Moiide, the computation of Jc is 

log2 7r 0.7981798684 

Compl. logj{ r.4374022154 

Compl. log v/(l + /i) . . . 9.9999993882 

\ogJc 8.2355814720 

1c = 0.01720210016. 

11. 

The following method of solving the equation 
M=E—ewiE, 
is recommended by Encke, Berliner Astronomisches Jahrhich, 1838. 
Take any approximate value of E, as e, and compute 

M' = £ — /' sin e , 



* The numbering of the Notes of the Appendix designates the articles of the original work to 
which they pertain. 

(279) 



280 APPENDIX. 

e" being used to denote e expressed in seconds, then we have 

or 

M— 3f = E—t — e" (sin E— sin e ) 
= {E—e) (1 — ecose), 

if ^ — e is regarded as a small quantity of the first order, and quantities of 
the second order are neglected for the present : — so that the correction of g is 

M—3f' 

x^ , 

1 — e cos £ 

and a new approximate value of e is 

M— M' 
"• 1 — e cos £ ' 

with which we may proceed in the same manner until the true value is obtained. 
It is almost always unnecessary to repeat the calculation of 1 — e cos e. Gener- 
ally, if the first s is not too far from the truth, the first computed value of 
1 — e cos g may be retained in all the trials. 

This process is identical with that of article 11, for X is nothing more than 
. d log sin E cos E 

dE ~sir^' 

if we neglect the modulus of Briggs's system of logarithms, which would subse- 
quently disappear of itself, and 

_ d\o^{e"imE) _ 1 
^ d{^'&mE) esin^' 

therefore, 

^ — l l — eco^E^ 
and 

x = ^{M-\-e'mit — ^) = {31— M') -^. = ^ ^~^^', , 

II + 1^ ' ' ^ /<+^ 1 — ecos^' 

and the double sign is to be used in such a way that X shall always have the same 
sign as cos E. In the first approximations when the value of g differs so much 
from E that the differences of the logarithms are uncertaui, the method of this 
note will be found most convenient. But when it is desired to insure perfect 
agreement to the last decimal place, that of article 11 may be used with 
advantage. 



APPENDIX. 



281 



As an illustration, take the data of the example in article 13 
Assume e = 326°, and we find 
log sine 9.74756 w 
log d' 4.70415 
log /'sine 4.45171 w 

d' sin e = — 28295" = — 7° 51' 35" 
Jf' = £ — e" sin e = 333° 51' 35" 
Jf— if' = — 4960" 

^~^' = — 6226" 

1 — e cos 6 

And for a second approximation, 

£ = 326° — 1° 43' 46" = 324° 16' 14' 

log sine 9.7663820?« 
log /' 4.7041513 
log/' sin £ 4.4705333 w 
e"sin£ = — 29548".36=- 
M' = 332° 28' 42".36 
if— Jf' = 4-12".41 

f 15".50 



log cos e 


9.91857 


lege 


9.38973 


log e cos e 


9.30830 


1 — 6 cose =.79662 


log (1 — ecose) 


9.90125 


logjf— Jf' 


3.69548W 


, M—M' 


3.79423m 



12' 28".36 

log(l — ecose) 
log(Jf— if') 



M—M' 



1 — e cos £ 

which gives 



, M—M' 

lOff- 

° 1 — e cos i 



Ez= 324° 16' 14" -f 15".50 = 324° 16'29".50. 



9.90356 
1.09377 

1.19021 



Putting 



we have 



18. 

g- = I JO = perihelion distance, 

x=:^y/^, 
log x = 8.0850664436, 

. tan hv-\-k tan^ \v^=.v.'c, 
T = — (3 tan ^ y -[- tan^ \ v) j 
36 



282 APPENDIX. 

a table may be computed from this formula, giving v for values of t as the argu- 
ment, which will readily furnish the true anomaly corresponding to any time 
from the perihelion passage. Table Ha is such a table. It is taken from the 
first volume of Annales de V Ohservatoire Imperiale de Paris, (Paris, 1855,) and difiers 
from that given in Delambre's Astronomy, (Paris, 1814,) Vol. III., only in the 
intervals of the argument, the coefficients for interpolation, and the value of ^ 
with which it was computed. 

The true anomaly corresponding to any value of the argument is found by 
the formula 

V = vo -{-A^ir— To) -^-A^it — T^f + (t — x^f ^3 -\-Ai{'v — Xq)\ 

The signs of A-^, A^, A^, are placed before the logarithms of these quantities 
in the table. 

Burckhabdt's table, Bowditch's Appendix to the third volume of the Mecanique 
Celeste, is similar, except that log r is the argument instead of t. 

Table Ila contains the true anomaly corresponding to the time from peri- 
helion passage in a parabola, the perihelion distance of which is equal to the 
earth's mean distance from the sun, and the mass ^ equal to zero. For if we put 
^=: 1, jit = 0, we have t^t. 

By substituting the value of x in the equation 

T = — (3 tan 2 V -\- tan^ h v) 

it becomes 

T= 27.40389544 (3 tan ^ v + tan^ i v) 

= 1.096155816 (75 tan ^y + 25 tan^iyj 
and therefore, if we put )c'= 0.912279061, 

75 tan ^ y -|- 25 tan^ ^v='/t 
log >«' = 9.9601277069 
Barker's Table, explained in article 19, contains x' r for the argument v. 
The Mean daily motion or the quantity M, therefore, of Barker's Table may be 
obtained from table Iln, for any value of v, by multiplying the corresponding 
value of T by v! . 

The following examples will serve to illustrate the use of the table. 
Given, the perihehon distance ^ ^ 0.1 ; the time after perihelion passage 
if^ 6'^590997, to find the true anomaly. 



APPENDIX. 283 

Assuming ^it = 0, we find 

7 = 208.42561 
To = 200. 
T— To= 8.42561 
^0 = 110° 24' 46'^69 
^(t — To)=+ri4'42".42 
A2{'v — rof = — 2'20'U9 
A,{<v — rof-=+ 4".76 

Ai{T — 'v,f = — 0".16 

i^ = IIP 3ri3".52 
or 

7=208.42561 

To =210. 
T — To = —1.57439 
vo = 111° 50' 16".87 
Ai{'r — T;o) = — 12'58".96 
^(t — To)2 = — 4".35 

As{T — tof = — 0".03 

A^{r — r,y = — 0".00 

t) = 111° 3ri3".53 
The latter form of calculation is to be preferred because the value of t — Tj, 
is smaller, and therefore the terms depending on (t — Tq), (t — r^f, (t — Tq)^, are 
smaller, and that depending on (t — To)* is insensible ; and it is the onlj form 
of which all the appreciable terms are to be found in the table. 

Beyond t = 40000, the limit of the table, we can use the formula, 

V = 180° — [6.0947259] Q* _ [6.87718] (~) — [7.313] Q*, etc., 

in which the coefl&cients expressed in arc are given by their logarithms. 
For T = 40.000, for example, we have 

«, = 180° — 10° 6' 6".87 — 3' 8".4 1 — 0".44 
= 169°50'44".28. 
If V is given, and it is required to find t, we have 



284 APPENDIX. 

For a first approximation the terms depending on the square and third power 
of 1 — To may be neglected, and the value of t — Tq thus found can be corrected 
so as to exactly satisfy the equation. 

If V exceeds 169°, the formula 

T = [1.9149336] tan ^ y -f [1.4378123] tan^ i v 

may be used instead of the table. 
Thus,fory = 169°50'44".28, 

logtan^K. .1.0513610 
1.9149336 



925.33 2.9662946 

logtan^^y. .3.1540830 

1.4378123 

39074.67 4.5918953 

T = 40000.00 

This method wiU often be found more convenient than the table, even where 
is less than 169°. 



35. 

Table Va contains Bessel's table here referred to, in a slightly modified 
form ; and also a similar table by Posselt, for the coefficients v' and v" in the 
formula of article 34, 

io = v-\-dv'-\-ddv" -{- d^ v'" + etc., 

it is taken from Encke's edition of Olbers AhJmndlung iiber die leicJdeste wid hequemste 
Mdliode die Bahn eims Cometen zu herechnen (Weimar, 1847). The following 
explanation of its construction and use is taken from the same work, with 
such changes as are needed to adapt it to the notation of the preceding 
articles : — 
K we put 

d- = tan h w 
T = tan i V 



APPENDIX. 285 

the formulas of article 34 become 



{i+xr 



1 (l+&y 

"• ', (1 + &^y " 

The second equation, in which v is expressed in terms of 2^, is that given by 
Bessel, Monatliche Correspondenz, Yol. XII., p. 197. He also gives the third coeffi- 
cient of the series, but has computed a table of only the first two. Posselt, in 
the Zeitschrift fur Astronomie und verwandte Wissenschaften, Vol. V., p. 161, has given 
the first equation ; he has also given three coefficients of the series, but a table of 
the second only, since Bessel's table wUl give the first coefficient simply by 
changing the sign. Posselt has changed the sign of the second coefficient also. 

Instead of the logarithms as given in the tables of Bessel and Posselt, the 
corresponding numbers are given in table Ya, and to avoid large numbers, 0.01 
is taken as the unit of (5". 

Patting 

tan i x = l 
the table contains 

^ — 10000 (i+a^ ^"^^^^ 

^ — 10000 (1 + r)* ^Ub^bb 

So that when r?; = ec we have 

y =: 2^ _|_ ^ ( 100 (5 ) -I- 5 ( 100 (^)2 
And when x=^v, 

^„=:y_^ (100 (^) — ^'(100(^)2 

It seems unnecessary to recompute the table in order to be certain of the 
accuracy of the last place, or to extend it further, as its use is limited. For 



286 APPENDIX. . 

absolute values of d greater than 0.03, and for values of x considerably greater 
than 90°, the terms here given would not be sufficient. In such cases the 
method of 37 and the following articles should be used. 
Example. — For Halley's comet, 

logd = 8.5099324, and ^ = 63''.43592, we have 

by table II«, to = 99° 36' 55".91 

and by table Va, A= i- 417.45 1st cor. + 22' 30".63 

5 = + 3.111 2d cor. 4- 32".57 

y=99°59'59".ll 

which, rigorously, should be 100° ; so that d is in this case too great. 
Inversely, we find, for v = 100°, 

v = 100° 0'00".00 
^=:^ 426.78 1st cor. — 23' 0".83 

B = -}- 0.297 2d cor. — rjl 

w= 99°36'56".06 

which agrees nearly with the preceding value. The change of the table to the 
present form has been made under the supervision of D'Arrest. 

39. 

When table Ha is used instead of Barker's table, w is the value of v, which 
corresponds to the argument 

at 



40. 



If we put 



^_ l-i.A-\-0 



the formulas for computing the true anomaly and radius vector are 

tan ^v= U„y tan i w 
r = E,q sec^ I v. 



APPENDIX. 287 

Table la for the Ellipse contains log ^^ and log^, for the argument ^, to- 
gether with the logarithms of their differences corresponding to a change of a unit 
in the seventh decimal place of the argument. It was computed by Prof J. S. 
Hubbard, and has been used by him for several years. Since it was in type, a 
similar table, computed by Mr. A. Marth, has appeared in the Astronomische Nach- 
richten, Vol. XLIII., p. 122. The example of article 43 will furnish an illustra- 
tion of its use. 

Formulas expressing the differentials of the true anomaly and radius vector 
in a very eccentric ellipse, in terms of the differentials of the time of perihelion 
passage, the perihelion distance and the eccentricity may be obtained from the 
equations of this article. 

If we put B =il, C= 0, we have, article 39, 



tan htv -\~ ^ tan^ ^ w =^ 



which, by article 20, gives 



dw « 7/ S at J , t J 

We also have, article 40, 

log tan ^ z; = log tsiu ^ w — Hog (1 — | ^ tan^ i to) -\- log y 
and, therefore, 

dv cos^^wdw _j_(^7 _|_ f -4 d^ 

2 sin 1 v COS i w 2 sin i w cos^ ^w (l — ^A) ' 7~ 1 —^A 'J' 

dv a cos^ ^ w _j 5 at cos^ ^w , 

%\nv~75i2,n^w{\—^A)"''' 2 ? 75 tan i ,<; (1 _ | ^) ^^ 

"I" 75 tan i w; (1 — I ^) ^^ I" y "T iZr|7 y 
which, by putting 

a cos^ i w 



K= 



75taniw;(l— J^) 



T— ^ 



M-. 



N= 



2(1 + 9.) 
4 



(l + e)(l + 9e) 



288 APPENDIX. 

0- ^^- 

p 



1-M 

10 



(l_e)(l + 9e) 

is reduced to 

ll^^ — KdT—KLtdq-\-\_KMt — ]Sr—OP^de, 

observing that dt-=. — dT, if T denotes tlie time of perihelion passage. 
If we differentiate the equation 



we find 



1 -)- e cos 1> 






These formulas are given by Nicolai, {Moncdliche Correspondenz, Vol. XXVH, 
p. 212). The labor of using them is greatly abridged by the fact that K, L, 
M, etc., are computed once for all, and that the quantities needed for this pur- 
pose are those required for computing the true anomaly and radius vector. 

If the ellipse so nearly approaches the parabola that, in the coefficients, we 
may assume 

tan hv=.y tan h w 

jp- ^ \/ 2 cos" \ V 

2 q^i tan \ v 

the values oi dv and dr assume a much more simple form. In this case we 
should have 

Tr- • h^2 cos^ \vsm.\v ^ y/ 2 cos* ^v k\j2q 

~ 2^1 tan it; ~ ^t ~~ ^* 

r TKT \ n ■n\ • \ 4 20tan*iw"l . 

_r 4-1-4 tan^^y ] . _ Stan^v 

— L(l-he)(l + 9e)J^''^^ — (l + «)(l + 9e) 



and consequently, 

'^^"■;^;^/^ + L-?^2(l + 9.)"~(l+«)(l+9e)J^^- 



J kU2q 



APPENDIX. 289 

This form is given by Encke {Berliner Astronomisches Jahr'buch^ 1822, page 184.) 
If we put g = 1 in the coefficient of c? e it becomes 

^— 20— ^2 5tan^^;. 

If we substitute the value of dv in the expression for dr given above, it 
may be reduced to the form 

dr:= =^= sin ^; t? r-f- cos vdq-{- (/q ■!!!!! 4- yL- r tan^ \v)de. 

Si'lq ^ sl2q ' 

41. 

The time t may be found from table Ila, by multiplying the value of r cor- 
responding to M' by 

^' B 



45. 

Table la, for the hyperbola is similar to that for the ellipse, and contains 
log E^ and log E, for the formulas 

tan \v ^=:^E^y tan h. w 
r = E^ sec^ hv. 

The differential formulas of article 40, of the Appendix, can be applied to 
the hyperbola also, by changing the sign of A and of 1 — e in the coefficients. 

56. 

As the solution here referred to may sometimes be found more convenient 
than the one given in articles 53-57, the formulas sufficient for the use of prac- 
tical computers are given below. 

Using the notation of 60 and the following articles, the expressions for the 
rectangular coordinates referred to the equator are, — 

a; = r cos u cos Q, — r sin m sin Q> cos i 
(1) ^ = r cos M sin 9, cos « -|- ^ sin u cos Q, cos ^ cos e — ^ sin u sin i sin e 
5! = r cos M sin Q, sin e -f- ^ sin u cos Q, cos i sin e -|- r sin u sin i cos £ 
37 



290 APPENDIX. 

which can be put in the form 

x = rsma sin [A -f- u) 

(2) ^ = rsmbsm{B -\- u) 

g = r sin c sin ( C -f- w) 
or 

a; = r sin a sin A cos u-{-r sin a cos A sin u 

(3) y = r sin b smB cos ^« -|- r sin b cos ^ sin m 
g = r sin c sin (7 cos u-{-r sin c cos Csin m 

equations (3), compared with (1) give 

sin a sm A=: cos Q, sin a cos J. = — sin Q cos e 

(4) sin ^ sin 5 ::= sin S2 cos £ sin J cos j5 = cos S2 cos ? cos e — sin ^' sine 
sin c sin C' = sin Q, sin e sin c cos C = cos Q cos ^ sin s -\- sin e cos 8 . 

By introducing the auxiliary angle E 



we shall find 



tan^— — ~ 
cos g^ 



cotan yl =z — tan g^ cos i 

tan gg cos ^ cos £ 
, /y cos i sin (^4- s) 

cotan C = — ^ — ^-7p^^ 

tan gg cos jfo sin £ 
•„ „ cosS^ sin Q cos ^ 

bill I* — : J- } 

Sin A cos A 

• 7 sin ^ cos £ COS Q, cos i cos £ — sin i sin e 

sm — — • "r> — — 75 

sm Is COS J} 



COS I sin £ -|- sin i cos e 



sin C COS O 



sin «, sin b, sine are always positive, and the quadrants in which A, B, C are to 
he taken, can be decided by means of equations (4). 

The following relations between these constants, easdy deducible from the 
foregoing, are added, and may be used as checks : 



, . sin 5 sin c sin ( C — B) 

sin a sin A 



APPENDIX. 291 

cos a = sin S sin i 

cos ^ = — cos Q, sin i cos e — cos / sin e 

cos c = — cos Q> sin i sin f -[~ cos / cos £ 

sin^ a -\- sin^ 5 -|- sin^ <? = 2 

cos^ a -|~ cos^ ^ H~ cos^ c ^ 1 
cos {A — B)=z — cotan a cotan h 
cos [B — 0)z= — cotan h cotan c 
cos (J. — ^) = — cotan a cotan c. 

58. 
K in the formulas of article 56 of the Appendix, the ecliptic is adopted as 
the fundamental plane, in which case £ = 5 and if we put 

n = long, of the perihelion 
sha.a=:kx A=zK^ — [n — ^) 
mi.l=^ky B = Ky — (tt — Q,) 
sinc = >^^ C=^K^ — (tt — Q,) 
we shall have 

Jc^ sin (^ — (tt — ^ )) = cos g? 
^^Cos(^ — (tt — S2)) = — sin g^ cos 2 
k^ sin ^ = cos g^ cos (tt — Q,) — sin g^ sin (tt — Q,) cos i 
haoB ^ = — [cos g^ sin (tt — Q>)-\- sin g^ cos (jt — 9,) cos 2] 
which can easily be reduced to the form, 

k^ sin K^ = cos^ i i cos n -\- dv? hi cos {n — 2 Q,) 
k^ cos K^=^ — [cos^ I i sin n -\- sin^ h ism {n — 2 $2 )] 
and in like manner we should find 

ky sin Ky = cos^ i i sin n — sin^ ^ z sin (tt — 2 9,) 
kyCosKy=^ cos^ ^2 cosu — sin^ h ^cos(7^ — 2 9) 

k^ sin K^ = sin i sin [tt — 9) 

k^cos K, = sin i sin (n — 9) 



292 APPENDIX. 

If these values are substituted in the general expression for coordinates, 

ak cos 9 CQ's, K woL E -\- ah sin K {cos JE — e) 
and if we put 

a cos (p = b 

a cos- iicosn 14- tan'' ^ ^ — ^^ 1 = A 

_^cos^ i ^• sinTc fl +tan2 ^ . sirK^r--2g)l ^^ 

L ' sin TT J 

acos^ ^ esin^r [l — tan^ |/ '^" ^^~'^^^1 = ^^ 

b cos^ I / cos 7t 1 — tan^ i i '^°'' ^^ ~ 1 = S 

L cos ;r J 

a sin ^ sin (tt — S2 ) = ^'^ 

^ sin i cos (jr — S^ ) = B" 

the coordinates will be 

x = A (cosE — g) + 5 sinE==^ (1 — esecE)^-^ sinE 
y = A[ (cos E — e) + ^' sin E = JL' (1 — e sec E) -j- B' sin E 
z = A" (cos E — e) 4- 5'' sin E = vl" (1 — e sec E) -f- 5" sin E. 

If the equator is adopted as the fundamental plane instead of the ecliptic, 
the same formulas may be used, if 9,, n, and i are referred to the equator by 
the method of article 55. Thus, if S^^ denote the right ascension of the node 
on the equator, for 9,, n, and i, we must use 9,^, 9,^-\-{n — 9) — -^ , and? 
respectively. 

This form has been given to the computation of coordinates by Prof Peirce, 
and is designed to be used with Zech's Tables of Addition and Subtraction Logarithms. 
Example. — The data, of the example of articles 56 and 58, furnish 
9 = 158° 30' 50".43, n = 122° 12' 23".55, i= 11° 43' 52".89 when the equator 
is adopted as the fundamental plane ; and also log b = 0.4288533. 
Whence we find 
log cos (tt — 2 a ) 9.9853041 n log sin (tt — 2 S2 ) 9.4079143 

log sec Tt 0.2732948 m logcosecTt 0.0725618 

logtan^^? 8.0234332 logtan^^e 8.0234332 

logc 8.2820321 - logc' 7.5039093 



add. log - 
log cos n 
log cos^ h i 
log a 

logJL 

add. log - 
log sin n 
log cos^ ^ i 
log h 

logB 



0.0082354 


C. sub. log - 


9.7267052 ?z 


log cos Jt 


9.9954404 


log cos^ i i 


0.4423790 


\ogb 


0.1727600 w 


logB' 


0.0013836 


C. sub. log -, 


9.9274382 


log sin n 


9.9954404 


log cos^ h i 


0.4288533 


log a 


0.3531155 w 


log^' 



293 



9.9916052 
9.7267052 
9.9954404 
0.4288533 
0.1426041 w 

9.9986120 

9.9274382 
9.9954404 
0.4423790 
0.3638696 



This method may also be used to compute Jc and K for the general formula 
of article 57. Thus: — 



add. log - 


0.0082354 


C. sub. log - 




9.9916052 


log cos TC 


9.7267052 w 


log cos TT 




9.7267052 ?2 


log cos^ k i 


9.9954404 


log cos^ i i 


' 


9.9954404 


log k^ sin Kj. 


9.7303810 ?? 


log ky cos Ky 




9.7137508 w 


add. log - 


0.0013836 


G. sub. log - 




9.9986120 


log sin n . 


9.9274382 


log sin TT 




9.9274382 


log cos^ i i 


9.9954404 


log cos^ i i . 




9.9954404 


log k^ cos K^ 


9.9242622^2 


log ky sin ^ 




9.9214906 


log tan K^ 


9.8061188 


log tan Ky 




0.2077398 J^ 


log cos K^ 


9.9254698 w 


log sin ^ 




9.9294058 


log^.=. 


= 9.9987924 




log ^^ = 


: 9.9920848 


K= 212' 


' 36' 56".l 


^,= 121° 4/28 


".1 



It will not be necessary to extend the example to the final expressions for 
x,y,z,9,% illustrations of similar apiDlications of the Addition and Subtraction 
Logarithms are given in the directions accompanying Zech's Tables. 



294 APPENDIX. 



If r, h, and I denote the radius vector, the heliocentric latitude and longitude 
of any planet, the rectangular coordinates referred to three axes, — of which 
that of X is directed towards the vernal equinox, that of z, parallel to the earth's 
axis, and that of y, 90° of right ascension in advance of x, — will be as in case 11. 

:r = r cos h cos I 

y =.r cos h sin I cos e — r sin 5 sin « 

= r cos b sin £ sin ^-j- r sin h cos £ 



and by putting 



cos b cos / 



sin h sin I cos h 

sm U = -.-— : = 



tan^_ . , 
sin ( 



sin d cos d 

tan b 



they assume the following forms convenient for computation : 

X = rcosu 

?/ = rsm u cos {6 -\- e) 

s = r sin u sin (^ -|- e) . 



74. 

The following are the solutions and examples from the Monatliche Correspon- 
denz referred to in this article, adopting the notation of article 74, and using L 
to denote the longitude of the Sun. 

Given, Q,, L',l,b, i, B, to find u,r, J, and the auxiliary angles A, B, 0, etc. 

L 

1. <^_^l(^-^l^ ^ tan A sinJ_tan (i^-^ ^ ^^^ ^^ 

sm (// — l) sin {A -\- t) 

cos (L' — Q) sm (i? + b) cos i 

o sin(L' — g2)tan5_ ^ sin C sin (Z' — g? ) _x 

4. cos(Z--S^)tan5^^^^^ sjn^ an (L'-^ ) cos (Z^ - Z) ^ ^^^ ^^ 

cos (Z — Z) tan ^ sin (Z) -}- Z — /) cos i 



APPENDIX. 295 

The angle to is to be taken between 0° and 180° when b is positive, and be- 
tween 180° and 360° when b is negative. When b = 0°, the body is in one of the 
nodes of its orbit, in the ascending node when sin (X' — I) and sin [l — Q.) have 
the same sign ; and in the descending node when they have opposite signs. 

It is immaterial in which of the two quadrants that give the same tangent, 
the auxiliary angles A, B , C, etc., are taken. In the following examples they 
are always taken between -\- 90° and — 90". 



II. 



5. -^^, — T— = tan U 



sin ^ sin 



{L'-Q)_ r 



sin {I — Q ) sin (i — E) sin u R 

a J. • • ti r^\ 4. rr cos^sin(X' — g2)sin6 r 

6. tan z sm (I — Q) = tSinF . ,„ \. . °°^ — ^ = -= 

"^ ^ sm i^M — 0) sin M cos ^ Ji 

^ , j_ • n cos (t sin (L' — T) r 

7. cos I tan u = tan 6r -;— 77 — ^ ^. — = ^ 

sin (f — Q, — Cr) cos M Ji 

Q tan (I — Q,) , TT • sin ^sin (L' — I) r 

O. : = tan jtZ —. — ;rYr ^i — ^ — /? /=r\ ^= "n 

cos » Sin (JI — u) sm{l — g^ ) Ji 

9. . . ^7/ ^, tan/ sjnlcosil^r 

sln^cos(< — hi) sm{u — J) Ji 

T n • • / 7 ^ \ i J T?- COS ^sin b cos (L' — Q ) r 

10. sm 2 cos (^ — Q) tan u = tan K r-r^ — 7^ ^^ = s 

^ ' sin (A — 0) cos u R 

-| -| sin C sin (Z — I) , j sin Z r 

cos {G-\-L' — l) tan {L! — Q,) cos%~^^'^ ^ sin {u—L) cos (Z'— ^ ) ~~ ^S 

1 sini)cos(Z^ — a) _ ^ sin^¥ r 

cos {p-\-L'— ^ )cos i ~ "^^ ^" sin {u — M) cos (Z' _ ^) — i? 



III. 



sm 



-I ^ R sin Z^sin (Z^ — g^ ) sin z R cos E sin (Z' — g^ ) sin i . 

sin (i — Z^)sin6 sin {i — E) sin {I — g^ ) cos 5 

-I t- ^cosi^sin(Z' — g^ ) tan i R sin J' sin (Z' — g^ ) sin (/ — g^ ) . 

sin {F— b) ~ sin {F—b) — ^ 

Other expressions for J may be obtained by combining 13 with all the 
formulas II. 

Examples : — 

Given, 9>= 80°59'12".07, r=:281°r34".99, ^=53°23'2".46, i= 10°3r9".55, 
/; = — 3° 6' 33".561, log i^= 9.9926158. 



296 



APPENDIX. 



log tan h 
log cos [L' — Q.) 
C\ogsm{L' — b) 
log tan A 

^ = — 3° 57' 
A-^i-- 



8.7349698 « 
9.9728762 n 
0.1313827 n 
8.8392287 ?e 
2'M36 
6° 40' 7''.414 



log sin A 
log tan [L' — ^ 
dog sin {A-\-i) 
loo* tan u 



8.8381955« 

9.5620014 

0.9350608 



9.3352577 ?i 
^(^_12°12'37".942 



log sin [L' — /) 
log tan i 
67. log cos (i/ — 
log tan B 



9.8686173 « 

9.2729872 

0.0271238^2 



9.1687283 
^ = 8°23'2r.888 
^-f ^ = 5°16'48".327 



log cos B 
log sin h 

logtan(X' — S2) 
<7. log sin (5 -I- 5) 



C. log cos i 



log tan i 



9.9953277 
8.7343300 » 
9.5620014 
1.0360961 
0.0075025 
9.3352577 w 



I) 



log sin(Z' — 
log tan h 
a. log sin [L' - 
C. log tan i 
log tan C 

c= — r 

0-\-L' — 9,== 192 



9.5348776 « 
8.7349698 w 
0.1313827W 
0.7270128 
9.1282429 « 
39' 7".058 
23' 15".864 



log sin C 

log sin {L' — Q,) 

C.logsin(C+i'- 

C. log cos i 

loff tan u 



9.1243583 n 
9.5348776 n 
) 0.6685194/2 
0.0075025 
9.3352578 w 



log cos (Z' — a ) 
log tan b 

67. log cos (X'—/) 
C. log tan i 



log tan D 



9.9728762 « 
8.7349698;? 
0.1714973/2 
0.7270128 



9.6063561 w 
i? = — 2r59'51".182 



log sin D 

log tan [L' — 9,) 

log cos (X' — I) 

C.logsin(i> + X'- 

C. log cos i 

log; tan u 



9.5735295 n 
9.5620014 
9.8285027 « 
■/) 0.3637217 w 
0.0075025 
9.3352578 « 



D^L' — l^ 205°38'41".348 



APPENDIX. 

5°. 



297 



log tan h 
log sin (/- 
log tan E 



8.7349698 w 
9.6658973 w 



9.0690725 
6°4ri2".412 
3° 55' 57".138 



log sin E 

log sin (Z'— £2) 
(7. log sin (e — E) 
C. log sin u 

0gr=::l0gi? + l0g^: 



9.0661081 

9.5348776 w 
1.1637907 
0.6746802 « 
0.4394566 
0.4320724 



log tan i 
log sin (/ — 
log tan F 

F—h = 







9.2729872 
9.6658973 ;z 



8.9388845 w 
•4°5r53".955 
r5r20".394 



log cos F 

log sin h 

log sin (Z' — U) 

a\ogmi{F—h) 

C. log sin u 

C. log cos i 



9.9983674 
8.7343300 w 
9.5348776 ?2 
1.4896990 7^ 
0.6746802 w 
0.0075025 w 
0.4394567 



log cos % 
log tan u 
log tan G 



9.9924975 
9.3352577 w 



9.3277552^ 
G^ = — 12° 0'27M18 



log cos G 9.9903922 

log sin (Z' — ^) 9.8686173 72 

aiogsin(/— ga— 6^) 0.5705092 7^ 

C: log cos M 0.0099379 



■S2—G^ = — 15° 35' 42^.492 


^^g^ 


0.4394566 


8°. 
logtan(/— $2) 9.7183744?2 logsin^ 


9.6717672 7z 


log cos ^■ 9.9924975 


log sin (Z' — /) 


9.8686173 w 


log tan ^ 9.7258769 72 


C. log sin {H — u) 


0.564969572 


H=—2^° 0'39".879 


C.\og^m{l — 9,) 


0.334102772 


^— «=— 15°48' r.937 


l^gi 


0.4394567 



38 



298 



APPENDIX. 



log tan h 


8.7349698 n 


log sin / 




9.4991749^ 


C. log sin i 


0.7345153 


log sin {L' - 


-9) 


9.9728762 » 


(7. log cos {I— 


-9,) 0.0542771 


G. log sin [u — 


-^) 


0.9674054 


log tan / 


9.5237622 w 


1°S5 




0.4394565 


/=- 


- 18° 23' 55".334 








u — I = 


6° ir 17".392 


10°. 






log sin i 


9.2654847 


log cos K 




9.9997290 


logcos(/ — 


-9,) 9.9475229 


log sin h 




8.7343300 w 


log tan u 


9.3352577 ?2 


log cos [L' - 


-9) 


9.9728762 w 


log tan K 


8.5482653 w 


0. log sin {K- 


^h) 


1.7225836 


K= 


= — 2°r26".344 


C. log cos u 




0.0099379 


K—b^ 


= Vb' 7".217 


i°si 




0.4394567 



log sin C 

log sin [L' — /) 
C.logcos(C-f-r— /) 0.1156850 ?e 
6'.logtan(i:'— a) 0.4379986 
a log cos/ 0.0075025 

log tan Z 9.5541617 n 

i: = — 19° 42' 32^.533 

u — L= 7°29'54".591 

12°. 

DJ^L'—9,=^M^° 2' 3r'.738 

log sin Z> 9.5735295 ?i 

logcos(Z' — S2) 9.9728762 n 

^.logcos(Z>+X'—S) 0.0002536 « 

CAogcoBi 0.0075025 

logtanil/(=X) 9.5541618 w 



11°. 

C + r — /=219°59'25".474 
9.1243583 ^^ 
9.8686173 « 



log sin L 
0. log sin [u — L) 
{7.log cos (i' — 9> 



logr 
log sin u 
log sin i 
C'. log sin 5 
loo;// 



13° 



9.5279439 w 

0.8843888 
0.0271238 » 
0.4394565 



0.4320724 
9.3253198 m 
9.2654847 
L2656700W 
02885469 



APPENDIX. 299 

76. 

If in the equations of article 60, 

X — X = // cos d cos a 
y — F=//cos^sina 
z — Z=. J Bind 

a denoting the right ascension, and d the decHnation, we suppose X, Y, Z known, 

we have 

dx=^ cos a cos d d J — z/ sin a cos d da — J cos a sin d dd 
d y = &hia cos d d J -]- J cos cc cos d da — J sina smd dd 
d s = smd dJ-\-/l cos d dd. 

Multiply the first of these by sin a , and subtract from it the second multiplied by 

cos a , and we find 

// cos d da = — dxwia -\~ dy sin a . 

Multiply the first by cos a and add to it the second multiphed by sin a , and 

we find 

dx cos a -{-dy sin a = cosd d J — J sin d dd. 

Multiply this equation by — sin d and add it to the third of the differential equar 
tions above multiplied by cos d and we find 

— dx cos « sin (^ — dy sin a sin ^ -|- ds cos d =J dd 
and, therefore, 

r> , sin a ^ , cos a -, 

COS «t a = — dx -\ J- dy 

-, rv COS a. sin 5 , sin a sin 5 , , cos 5 , 

d8 = ^ dx ^ — dy^-j-dz. 

From the formulas of article 56 of the Appendix are obtained 

dx X dy y dz z 

dr r' dr ?• ' dr r' 

—^■=x cotan (^ + m) , ^ =y cotan {B^u), -£ = 2 cotan ( 6^+ u) 

dx . dv . , dz . 

-— =: a; sm ^( COS « , -^=:r smw coso, ^-. = r smwcos e, 

di ^ di ' a% ' 

and the partial differentials 

dx . dy dz . 

jT-== — ycose — ssme, ^ = a; cose, — ^i?;sm6 



300 APPENDIX. 

whence 

dx=^-dr -j-^cotan [A -\-u) c? y -|- a; cotan ( J. -|- ?<) dn 

— [x cotan {A -\- u) -\- y 0,0?, B -\- z Bux t] d Q, -\- r miu c(i& a di 

dy = ^-dr -\- y cotan (^-f-^) dv -\-y Q.oi-d,n [B -\-ii) dn 

— \_y cotan {B -\-u) — x cos i\d9,-\-r^\r\. u cos h di 

dz = -dr — z cotan [C -\-u) dv -\- z cotan [C -\- u) d tc 

— [z cotan ( C-\- u) — xwa.B]d Q,-\- r sin u cos c di. 

These formulas, as well as those of 56 may be found in a small treatise 
Ueler die Differeniialformeln fur Comete^i-Bahnen, etc., by G. D. E. Weyer, (Berlin, 
1852). They are from Bessel's Abhandhng iiher den Olbers'schen Cometen. 

90. 

Gauss, in the Berlimr Astronomisches Jakrbuch for 1814, p. 256, has given an- 
other method of computing \, and also C of article 100. It is as follows : — 
We have 

^~"** 6 ' 9X— X 

This fraction, by substituting for X. the series of article 90, is readily trans- 
formed into 

t_ 8 „/. , 2.8 . 3.8.10 . . 4.8.10.12 5.8. 10 . 12. 14 , \ 

^~105^V^ + ~9"^+ 9.11 ^+ 9.11.13 -^ + 9.11.13.15 ^ "T ^tC. j 

Therefore, if we put 



we shall have 



^-l + ^gi^^ + i|^V + etc., 



xX-\X^^- = -^\^A:^ 



by means of which ^ can always be found easily and accurately. 



APPENDIX. 301 

For C, article 100, it is only necessary to write z in place of x in the pre- 
ceding formulas. 

A may be computed more conveniently by the following formula : — 

A_[i. — x) V^i-2.9^-r2.4.9.11^ + 2.4.6.9.11.13^ i" etc. j 



142. 

Prof. Encke, on the 13th of January, 1848, read a paper before the Eoyal 
Academy of Sciences at Berlin, entitled Ueier den Ausnahmefall einer doppelten 
Bahnbestimmung aus denselben drei geocentrischen Oertern, in which he entered into a 
full discussion of the origin of the ambiguous case here mentioned, and the 
manner in which it is to be explained. The following paragraphs, containing 
useful instructions to the practical computer, embody the results of his in- 
vestigation : — 

By putting 

5^ = ((y4-a), 

Equation IV., 141, becomes, for / > ^ 

m sin* z = sin [z — q) 
and for / <[ iZ' 

m sin* z = sin {z -\- q) 
m is always positive. 

The number and the limits of the roots of this equation may be found by 
examining both forms. 

Take the first form, and consider the curves, the equations of which are 

y -=. m sin* s, y' = sin {z — q) 

y and y' being ordinates, and z abscissas. 
The first differential coefficients are 

-f^im sin^ z cos z, -/- = cos [z — q\ 



302 appe:ndix. 

There will, therefore, be a contact of the curves when we have 

m sin* g = sin (^ — q) 
and 

4 m sin^ s cos z = cos (z — q) 
or when 

4 sin (s — q) cos s = cos (s — q) sin ^ 

which may be more simply written 

sin (2 s — q) = ^ sin q. 

When the value of s deduced from this equation satisfies 

m sin* ^ ^ sin (^ — q) 

then there is a contact of the curves, or the equation has two equal roots. These 
equal roots constitute the limits of possibility of intersection of the curves, or the 
limits of the real roots of the equation. 

For the delineation of both curves it is only necessary to regard values of 
s — q between 0° and 180°, since for values between 180° and 360° the solution 
is impossible; and beyond 360° these periods are repeated. 

The curve 

/ = sin (s — q) 

is the simple sine-curve, always on the positive side of ?/', and concave to the axis of 
abscissas, and has a maximum for 

z — q= 90°. 
The cm-ve 

^ = sin*0 

is of the fourth order, and since it gives 

—^ =z 4 «^ sin^ s cos ^ = msm2s — i m sin 4 s 
az 



3-5^ = 12 m sin^ s cos^ s — 4 m sin* s 



= 4 m sin^^ (1 4- 2 cos 2 2) = 2 m (cos 2s — cos 4 2) 
-7-4 = — 4 w (sin 2 g — 2 sin 4 s) 



dz* 

it has a maximum for 



- , = — 8 m (cos 2z — 4 cos 4 2) 

dz* ^ ' 



90° 



APPENDIX. 303 

and a point of contrary flexure for 

z = 60°, and z = 120°. 

From s = 0° to s = 60°, it is convex to the axis of abscissas, from 60° to 
120° it is concave, and convex from 120° to 180°. 
For osculation, the three equations, 

m sin* = sin [z — q) 
4 m sin^ z cosz= cos {z — g) 
4:msm^z{l -\-2 cos20)= — sm{z — q) 
must coexist, or 

m sin^z = sin (z — q) 
sin {2z — q) = ^sinq 
cos2s=: — |. 
In this case we should have 

sin (2 s — S') = I cos ^ -j- 1 ^i^ ^y 
consequently, 

tan q = ^ 
and 

sin^ = |, 
or 

g = 45° + isin-i|. 

From these considerations we infer that for the equation 

m sin* z = sin (z — q) 
or even when it is in the form 

m^ sin® z — 2 m cos q sin^ z -\- sin^ z — sin^ ^ = 

of the eighth degree, there can only be four real roots ; because, in the whole 
period from z — ^=0°to z — q = 360°,only four intersections of the two curves 
are possible on the positive side of the axis of ordinates. 

Of these, three are between 0=0° and 0^180°, and one between 180" 
and 180° -\- q ; or, inversely, one between 0° and 180°, and three between 180° 
and lSO°-\-q; consequently, there are three positive and one negative roots, or 
three negative and one positive roots for sin z. 



304 APPENDIX. 

Contact of the curves can exist only when for a given value of q, 

/ = iq-\-^ sin~^ I sin q 
and 

, sin (/ — g) 
sin* z 

If the contact of the curve of the fourth order with the sine-curve is with- 
out the latter, then will m constitute the upper limit, — for 771 greater than this 
values of the roots will be impossible. There would then remain only one jjositive 
and one negative root. 

If the contact is within the sine-curve, then will the corresponding ni" con- 
stitute the lower limit, and for m less than this, the roots again would be re- 
duced to two, one positive and one negative. 

If q is taken negative, or if we adopt the form 

m sin* 2 = sin [z -\- q) 

180° — z must be substituted for 0. 
The equation 

rr^ sin^ z — 2 w^ cos q sin^ z -\- sin^ z — sin^ q=^ 

shows, moreover, according to the rule of Descartes, that, of the four real 
roots three can be positive only when q, without regard to sigil is less than 
90°, because m is always regarded as positive. For q greater than 90°, there is 
always only one real positive root. Now since one real root must always cor- 
respond to the orbit of the Earth, that is, to r = JR' ; and since sin d', in the 
equation, article 141, — 

R' sins' 

sm z = 7 — 

r 

is always positive, so that it can be satisfied by none but positive values 
of z; an orbit can correspond to the observations only when three real roots are 
positive, or when q without regard to its sign is less than 90°. These limits are 
still more narrowly confined, because, also, there can be four real roots only 
when in lies between m' and 771", and when we have 

f sin ^ < 1, or sin ^ < f , ^ < 36° 52' 11".64 

in order that a real value of / may be possible. 



APPENDIX. 305 

Then the following are the conditions upon which it is possible to find a 
planet's orbit different from that of the earth, which shall satisfy three complete 
observations. 

First. The equation 

m sin* g = sin [z -\- q) 

must have four real roots. The conditions necessary for this are, that we must 
have, without regard to sign, 

sin^<| 

and m must lie between the limits m' and m". 

Second. Of these four real roots three must be positive and one negative. 

For this it is necessary that cos q should remain positive for all four of those 
values for which 

sin^<±|, 

the two in the second and third quadrants are excluded, and only values between 
— 36° 52' and + 36° 52' are to be retained. 

If both these conditions are satisfied, of the three real positive roots, one must 
always correspond to the Earth's orbit, and consequently will not satisfy the 
problem. And generally there will be no doubt which of the other two will 
give a solution of the problem. And since by the meaning of the symbols, arti- 
cles 139, 140, we have 

sin z sin {8' — z) sin §' 

not only must s and d' be always less than 180°, but, also, sin(d' — z) must be 
positive, or we must have 

d'>z. 

If, therefore, we arrange the three real positive roots in the order of their 
absolute magnitudes, there may be three distinct cases. Either the smallest root 
approaches most nearly the value of d', and corresponds, therefore, to the Earth's 
orbit, in which case the problem is impossible ; because the condition d' ^z can 
never be fulfilled. Or the middle root coincides with d', then will the problem 
be solved only by the smallest root. Or, finally, the greatest of the three roots 
differs least from d'. in which case the choice must lie between the two smaller 

39 



306 APPENDIX. 

roots. Each of these will give a planetary orbit, because eacli one fulfils all 
the conditions, and it will remain to be determined, from observations other than 
the three given ones, which is the true solution. 

As the value of m must lie between the two limits m and m", so also must 
all four of the roots lie between those roots as limits which correspond to m and 
m". In Table TV a. are found, therefore, for the argument q from degree to degree, 
the roots corresponding to the limits, arranged according to their magnitude, and 
distinguished by the symbols z^, z^, 0™, z". For every value of m which gives a 
possible solution, these roots will lie within the quantities given both for m and 
m", and we shall be enabled in tbis manner, if d' is found, to discern at the first 
glance, whether or not, for a given m and q, the paradoxical case of a double orbit 
can occur. It must, to be sure, be considered that, strictly speaking, d' would 
only agree exactly with one of the s's, when the corrections of P and Q belong- 
ing to the earth's orbit had been employed, and, therefore, a certain difference 
even beyond the extremest limit might be allowed, if the intervals of time should 
be very great. 

The root z", for which sin z is negative, always falls out, and is only intro- 
duced here for the sake of completeness. Both parts of this table might have 
been blended in one with the proviso of putting in the place of z its supplement ; 
for the sake of more rapid inspection, however, the two forms sin [z — q) and 
sin (.i -|- q) have been separated, so that q is always regarded as positive in the 
table. 

To explain the use of Table IVa. two cases are added ; one, the example of Ceres 
in this Appendix, and the other, the exceptional case that occurred to Dr. Gould, 
in his computation of the orbit of the fifth comet of the year 1847, an account of 
which is given in his Astronomical Journal, Vol. I., No. 19. 

I. In our example of Ceres, the final equation in the first hypothesis is 

[0.9112987] sin* .2 = sin {z — 7° 49' 2".0) 
and 

d' = 24° 19' 53".34 

the factor in brackets being the logarithm. By the table, the numerical factor 
lies between ni and m", and this d' answers to z", concerning which there can be 
no hesitation, since s"" must lie between 10° 27' and 87° 34'. Accordingly, we 



APPENDIX. 307 

have only to choose for the s' which occurs in this case, and which, as we per- 
ceive, is to be sought between 7° 50' and 10° 27'. 

The root is in fact 

2^=1" 59' 30".3, 
and the remaining roots, 

z^= 26 24 3 

, 0^ = 148 2 35 

0^=187 40 9 

are all found within the limits of the table. 

2. In the case of the fifth comet of 1847, Dr. Gould derived from his first 
hypothesis the equation 

[9.7021264] sm'z = sin {s + 32° 53' 28".5). 
He had also 

(J' = 133° 0' 31". 

Then we have sin g <C j, and the inspection of the table shows that the factor 
in the parenthesis lies between n/ and m" ; therefore, there will be four real roots, 
of which three will be positive. The given d' approximates here most nearly to 
2!™, about which, at any rate, there can be no doubt. 

Consequently, the paradoxical case of the determination of a double orbit 
occurs here, and the two possible values of s will lie between 

88° 29' — 105° 59' 
and 

105 59 —131 7 
In fact, the four roots are, 



s>= 95° 


31' 43".5 


g" = 117 


31 13 .1 


0- = 137 


38 16 .7 


0- = 329 


58 35 .5. 



By a small decrease of m without changing q, or by a small decrease of q 
without changing m, a point of osculation will be obtained corresponding to 
nearly a mean between the second and third roots; and on the contrary, by a 
small increase of m without changing q, or a small increase of q without changing 
m, a point of osculation is obtained corresponding to nearly a mean between the 
first and second roots. 



308 APPENDIX. 

We have, therefore, the choice between the two orbits. The root used bj Dr. 
Gould was 2°, which gave him an ellipse of very short period. The other obser- 
vations showed him that this was not the real orbit. M. D'Arrest was involved in 
a similar difficulty with the same comet, and arrived also at an ellipse. An ellipse 
of eighty-one years resulted from the use of the other root. 

" Finally, both forms of the table show that the exceptional case can never 
occur when d' < 63° 26'. 

" It will also seldom occur when d' < 90°. For then it can only take place 
with the first form sin (g — q), and since here for all values of q either the limits 
are very narrow, or one of the limits approximates very nearly to 90°, so it will 
be perceived that the case where there are two possible roots for d' < 90° will 
very seldom happen. For the smaller planets, therefore, which for the most part 
are discovered near opposition, there is rarely occasion to look at the table. For 
the comets we shall have more frequently d' > 90° ; still, even here, on account 
of the proximity to the sun, d' > 150° can, for the most part, be excluded. Con- 
sequently, it will be necessary, in order that the exceptional case should occur, 
that we should have in general, the combination of the conditions d' > 90° and 
q between 0° and 32° in the form sin (2 — q), or between 22° and 36° 52' in the 
form sin (s -j- q)." 



Professor Peirce has communicated to the American Academy several methods 
of exhibiting the geometrical construction of this celebrated equation, and of 
others which, like this, involve two parameters, some of which are novel and 
curious. In order to explain them, let us resume the fundamental equation, 

m sin^ s = sin {s — q). 
1. The first method of representation is by logarithmic curves ; the logarithm 
of the given equation is 

log m -|- 4 log sin s = log sin {s — q). 

If we construct the curve 

^ =z 4 log sin 0, 



APPENDIX. 309 

and also the same curve on another scale, in which y is reduced to one fourth of 

its value, so that 

y := log sin s, 

it is plain that if the second curve is removed parallel to itself by a distance equal 
to q in the direction of the axis of z, and by a distance equal to — log m in the 
direction of the axis of y, the value of z on the first curve where the two curves 
intersect each other will be a root of the given equation ; for, since the point of 
intersection is on the first curve^ its coordinates satisfy the equation, 

^ =: 4 log sin z, 
and because it is on the second curve its coordinates satisfy the equation, 

y -[- log m = log sin (s — q)\ 
and by eliminating y from these two equations we return to the original equation, 
m sin*0 = sin {z — q). 

A diagram constructed on this principle is illustrated by figure 5, and it will 
be readily seen how, by moving one curve upon the other, according to the 
changeable values of q and ?n, the points of intersection will be exhibited, and also 
the limits at which they become points of osculation. 

On this and all the succeeding diagrams, we may remark, once for all, that 
two cases are shown, one of which is the preceding example of the planet Ceres, 
in which the four roots of the equation will correspond in all the figures to the 
four points of intersection D, D', 17', D'", and the other of which is the very 
remarkable case that occurred to Dr. Gould, approaching the two limits of 
the osculation of the second order, the details of which are given in No. 19 of his 
Astronomical Journal, and the points of which are marked on all our diagrams 

G-, Cr , Cr , br . 

2. The second method of representation is by a fixed curve and straight line, 
as follows. 

[a.) The fundamental equation, developed in its second member, and divided 
by m Go&z, assumes the form 

sin* z cos <7 / , , V 

= — - ( tan z — tan a) 

cos 2 m ^ ^' 

By putting 

q, a 



X = tan z,b=z tan r " — ^ ^^ 



310 APPENDIX. 

the roots of the equation will correspond to the points of intersection of the 



with the straight line 

2/ = a (x — b). [Figs. 6 and 6'.] 

It will be perceived that the curve line, in this as in all the following cases 
under this form, is not affected hj any change iii the values of m and q, and that 
the position of the straight line is determined by its cutting the axis of x at 
the distance tan q from the origin, and the axis of ^ at the distance — ^^ 
from the origin. The tangent of its inclination to the axis is obviously equal to 
^^, which may in some cases answer more conveniently for determining its 
position than its intersection with the axis of ^. 

(b.) The development of the fundamental equation divided by m sin g, is 

sin^ s = -^ ( CO tan q — cotan s) ; 

and by putting 

X = cotan 2 

b = cotan g- 

m 

the roots of the equation correspond to the intersection of the curve 

y = sin^ s = (1 -\- x^)~^ 
with the straight line 

t/ = a{b — x). [Fig. 7.] 

The position of the straight line is determined by its cutting the axis of x at a 
distance equal to cotan q from the origin, and the axis of y at a distance equal to — ^ 
from the origin. This form of construction is identical with that given by M. 
Binet in the Journal de FEcole Poly technique^ 20 Cahier, Tome XIII. p. 285. His 
method of fixing the position of the straight line is not strictly accurate. This 
mode of representation is not surpassed by either of the others under this form. 

[c.) The fourth root of the fundamental equation developed, and divided by 
cos [z — q\ assumes the form 

i^mcos^(tan(« — q) ^ tan^?) = ^['os{z-q)^ - 



APPENDIX. 311 

By putting 

X = tan {z — q) 

b = tan q 

a =^ \l m cos q 

the roots of the equation correspond to the intersection of the curve 

_ ^ (sin {z — q)) ^ ^^ Q I ^^1 
^ cos (z — q) \ \^ ) 

with the straight line 

y = a{x^h). [Fig. 8.] 

The straight line cuts the axis of ;p at a distance equal to — tan q, and the axis 
of ^ at a distance equal to y' m sin q, from the origin. 

[d.) The development of the fourth root of the fundamental equation divided 

by sin {z — q) is, 

3 

'^ m mxq (cotan [z — q) -\- cotan q) = cosec (s — q). 

By putting 

^ = cotan {z — q) 

h = cotan q 

a = ^ m sin q 

the roots of the equation correspond to the intersection of the curve 

with the straight line 

^ = a{x-\-b). [Figs. 9 and 9'.] 

The straight line cuts the axis of a; at a distance equal to — cotan q, and the 
axis of ?/ at a distance equal to i^ m cos q, from the origin. 

{e.) From the reciprocal of the fundamental equation multiplied by m, its 
roots may be seen to correspond to the intersection of the curve 

r = cosec^ z 
with the straight line 

r = m cosec [z — q). [Figs. 10 and 10'.] 

Both these equations are referred to polar coordinates, of which r is the radius 
vector, z the angle which the radius vector makes with the polar axis, m the dis- 
tance of the straight line from the origin, and q the inclination of the line to the 
polar axis. 



312 APPENDIX. 

(/). From tlie reciprocal of the fourth root of the fundamental equation, its 
roots may be seen to correspond to the intersection of the curve 



with the straight line 



which 



r = cosec* (jp 
= ^-cosec(y4-^), 
--z-q. [Fig. 11.] 



Both these equations are referred to polar coordinates, of which y is the 
angle which the radius vector r makes with the polar axis, y/ - the distance of the 
straight line from the origin, and q the inclination of the line to the polar axis. 

3. The tliird method of representation is by a curve and a circle. 

(«.) The roots of the fundamental equation correspond to the intersection 
of the curve 



with the circle 



^=^sin(^-4 [Fig. 12.]. 



Both these equations are referred to polar coordinates, of which r is the radius 
vector, z the ano-le which the radius vector makes with the polar axis, — the 
radius of the circle which passes through the origin, and 90° -\- q \% the angle 
which the diameter drawn to the origin makes with the polar axis. 

(^.) From the fourth root of the fundamental equation it appears that its 
roots correspond to the intersection of the equation 

r = y' sin g) 
with the circle 

r=^ m sin (9 -f q) [Fig. 13], 

in which (p^{z — q) is the inclination of the radius vector to the polar axis, 
^m is the diameter of the circle which passes through the origin, and 90° — q 
is the inclination of the diameter drawn through the origin of the polar axis. 

In these last two delineations the curve IK I' K' I" incloses a space, within 
which the centre of the circle must be contained, in order that there should be 
four real roots, and therefore that there should be a possible orbit. The curve 



APPENDIX. 



313 



itself corresponds to tlie limiting points of osculation denoted by Professor Encke's 
m' and vd', and the points K and K' correspond to the extreme points of oscula- 
tion of the second order, for which Encke has given the values 5' = =f 36° 52' 
and w/ = 4.2976, and m' = 9.9999. 

On the delineations, /S is the centre of the circle for our example of Ceres, 
and /S' the same for Dr. Gould's exceptional case. A careful examination of the 
singular position of the point /S" will illustrate the peculiar difficulties attending 
the solution of this rare example. 

159. 

We add another example, which was prepared with great care to illustrate the 
Method of Computing an Orbit from three observations published in pamphlet 
form for the use of the American Ephemeris and Nautical Almanac ixi 1852. It 
furnishes an illustration of the case of the determination of two orbits from the 
same three geocentric places, referred to in article 142. 

We take the following observations, made at the Greenwich Observatory, 
from the volume for the year 1845, p. 36. 



Mean Time, Greenwich. 


Apparent Eight Ascension. 


Apparent Declination. 


1845. July 30, Ti 5 lols 
Sept. 6, 11 5 56.8 
Oct. 14, 8 19 35.9 


33°9 51 15J5 
3.S2 22 39.30 
328 7 51.45 


,S'. 23 31 34.60 
27 10 23.13 
26 49 57.23 



From the Naidical Almanac for the same year, we obtain 



Longitude of the Sun 
from App. Equinox. 



Distance from the 
Earth. 



Latitude of the 
Sun. 



Apparent Obliquity 
of the Ecliptic. 



July 30. 
Sept. 6. 
Oct. 14. 



127 40 11.32 
164 9 40.85 
201 21 12.49 



-14.99 
-14.06 
-12.16 



0.0064168 
0.0031096 
9.9984688 



—0.17 
+0.21 
-1-0.53 



27 28.13 

28.41 
28.05 



The computation is arranged as if the orbit were wholly unknown, on which 
account we are not at liberty to free the places of Ceres from parallax, but must 
transfer it to the places of the earth. 

40 



!14 APPENDKL 

Reducing the observed places of the planet from the equator to the ecliptic, 
ve find 



Date. 


App. Longitude of Ceres. 


App. Latitude of Cores. 


July 30. 
Sept. 6. 
Oct. 14. 


332 28 28.02 
324 35 58.87 
321 4 54.55 


S. 13 54 52.47 
14 45 30.00 
13 5 35.33 



And also. 



Date. 


Longitude of Zenith. 


Latitude of Zenith. 


July 30. 
Sept. 6. 
Oct. 14. 


11 6 
4 49 
1 4 


K 53 26 
56 22 

58 4 



The method of article 72 gives 



Date. 


Reduction of Longitude. 


Reduction of Distance. 


Reduction of Time. 


July 30. 
Sept. 6. 
Oct. 14. 


+16.32 
- 7.10 
—26.95 


+0.0001368 
1421 
0907 


—0.070 
—0.065 
—0.071 



The reduction of time is merely added to show that it is wholly insensible. 

All the longitudes, both of the planet and of the earth, are to be reduced to 
the mean vernal equinox for the beginning of the year 1845, which is taken as 
the epoch ; the nutation, therefore, being applied, we are still to subtract the 
precession, which for the three observations is 28''.99, 34".20, and 39".41, re- 
spectively; so that for the first observation it is necessary to add — 43"'.98, for 
the second, — 48".26, and for the third, — 5r'.57. 

Finally, the latitudes and longitudes of Ceres are to be freed from the aber- 
ration of the fixed stars, by subtracting from the longitudes 18".76, 19".69, and 
10".40, respectively, and adding to the latitudes — 2.02, +1.72, and -|-4.02, 
numbers which are obtained from the following formulas of Prof Peirce : — 

d a = m cos (O — a) sec {i 

(J /:? = msin (© — a) sin a ; 

where O =: sun's longitude, and m ■= aberration of ©. 



APPENDIX. 



!15 



The longitudes of the sun were corrected for aberration by adding 20".06, 
20''.21, and 20".43, respectively, to the numbers given in the Nautical 
Almanac. 

These reductions having been made, the correct data of the problem are as 
follows : — 

Times of observation. 



For Washington Meridian. 
Ceres'slong. a, «', a" 
latitudes /:J, {i', f 
Earth's long. I, I', l" 
loffs. of dist. R, R', R" 



July 30. 372903. 

330° 27' 25".28 
— 13 54 54 .49 
307 39 43 .66 
0.0064753 



Sept. 6. 248435. 

324 34 50.92 
— 14 45 28.28 
344 8 45.49 
0.0031709 



By the formulas of Arts. 136 and 137, we find 



7,7,7 ' ' • . 
d,d',d" . . . . 
log d, d', d" sines 
AD,AD\AD" 
A'D,A:'iy,AD", 



I rr 



log £, e', b" sines, 
log sin h «' 
log cos ^ e' 



329° 25' 34".81 
28 12 56 .84 

9.6746717 

199° 45' 41".00 

233 54 11 .72 

27 32 45 .72 

9.6650753 



218° 11' 22".38 

24 19 53 .34 

9.6149131 

204° 8'25".14 

233 31 23 .54 

142 37 25 .44 

9.7832221 

9.9764767 

9.5057163 



And by article 138, 

log^sin?; .... 6.2654993 w 

logTco^t 9.2956278?? 
wherefore 

t=im° 3' 12".63, log ^ . . . 9.2956280 

t-\-7'= 38°14'35".01, logsin(^ + /) 9.7916898 

log /S' 8.6990834 

log ^ sin (^-f-/) . . 9.0873178 ' 

Whence log tan ((^' — 0) . . . 9.6117656 

d' — G = 22° 14' 47".47 and a = 2° 5' 5''.87. 



Oct. 14. 132915. 

321 3 52.58 
— 13 5 31.31 
21 19 53.97 
9.9985083 



194 69 35 .15 

61 6 50.78 

9.9422976 

203° 56' 46".56 

199 30 24 .04 

115 4 41 .10 

9.956992 



316 APPENDIX. 

By articles 140-143, we find 

A" D' — d" = 172° 24' 32".76 log sin 9.1208995 log cos 9.9961773 n 
AD' — 8 =175 55 28.30 8.8516890 9.9989004« 

A:'D — d" =172 47 20.94 9.0987168 

AD — d'^o =177 30 53.53 8.6370904 

Aiy' — d =175 43 49.72 8.8718546 

A:i)" — d'-^o=m 15 36.57 8.6794373 

log a 0.0095516, a =1.0222370 

log 5 0.1389045. 

Formula 13, which serves as a check, would give log i = 0.1389059. We 
prefer the latter value, because sin {A D — id' -|- a) is less than sin {A U' 
-8'J^o). 

The interval of- the time (not corrected) between the second and third obser- 
vations is 37.884480 days, and between the first and second 37.875532 days. 
The logarithms of these numbers are 1.5784613 and 1.5783587 ; the logarithm 
of k is 8.2355814 ; whence log ^ = 9.8140427, log r = 9.8139401. 

We shall put, therefore, for the first hypothesis 

2: = logP = ^" =9.9998974 

^ = log ^ = (3 r = 9.6269828 
and we find 

w = 5° 43' 56".13 
w4-a = 7 49 2 .00 
log Q c sin M = 0.9112987 

It is found, by a few trials, that the equation 

^ c sin ca sin* s = sin {z -j- T 49' 2".00) 

is satisfied by the value 

z = T 59' 30".30, 

whence log sin z = 9.1431101, and 

r'=^?^'= 0.474939. 



APPENDIX. 317 

Besides this solution, the equation admits of three others, — 

z= 26° 24' 3" 
^=148 2 35 
g=187 40 9 

The third must be rejected, because sin z is negative ; the second, because z is 
greater than d' ; the first answers to the approximation to the orbit of the earth, 
of which we have spoken in article 142 * 

The manner of making these trials is as follows. On looking at the table of 
sines we are led to take for a first approximation for one of the values, gr = 8" 
nearly, or 8° -}- x. Then we have 

log sins 9.14356 + 89 a; 

logsm*0 6.57424 + 356 :» 

log ^csin to 0.91130 

logsin(0 — w — (j) . . . 7.48554-1- 356 iP 

g_a,_a = 0°10'52''-f-^\V9^ 
w-l-(T = 7 49 3 

2=7 59 55 -f- 1^2 ^j nearly = 8° -|- a?. 
For the second approximation, we make 

z—T 59' 30'' -f- x ; and have 

log sins 9.1431056 + 150 a;^ 

log sin* s 6.5724224 + 600 «^, 

^csinw . . . . . 0.9112987 
log sin (s — o — (t) . 7.4837211 + 600 a;' 
g _ w — (T = 0° 10' 28".27 + iV ^' nearly. 
w + (7 = 7 49 2. 00 

z=n 59 30. 27 + iV^'='J'°59'30".30. 
The process is the same for the other roots. 

* See article 142 of the Appendix. 



318 APPENDIX. 

Again, by art. 143 we obtain 

C= 185^0' 3r.78 

r = 189 25 30 .25 

log r= 0.4749722 

log /'=: 0.4744748 

i (^" _|_ u) = 264° 2V 48''.61 

^u'' — u) = 2SS 49 5.19 

2/ = 6 57 7 .46 

2/" == 6 56 32 .68 

The sum 2/-|-2/", which is a check, only differs by 0"'.20 from 2/', and the 
equation 

p __ r sin 2f" __ ?/' 
/' sin 2/ w 

is sufficiently satisfied by distributing this 0".2 equally between 2/ and 2/", so 
that 2/= 6°59'7".36, and 2/"= 6°56'32".58. 

Now, in order that the times may be corrected for aberration, the distances 
q, q', q" must be computed by the formulas of Art. 145, and then multipHed into 
the time 493^ or 0^005706, as follows: — 

logr 0.47497 

logsin(AD — Q .... 9.51187 
comp. log sin ^ 0.32533 



log 9 




0.31217 


log const 




7.76054 


log of reduction 




8.07271 


Eeduction = 


: 0.011823 




log/, 




0.47497 


log sin [d — z) 




9.44921 


comp log sin d', 




0.38509 


log of reduction 




0.30927 


Reduction, 0.011744. 





The constant of aberration is that of M. Struve. 





APPENDIX. 






log/' 0.47447 






logsm(A''D' — n . . . 9.84253 






log sin r 0.05770 






log of reduction .... 0.37470 






Reductions 0.013653 




Observations. 


Con-ected Times. Intervals. 


Logarithms. 


I 


July 30. 361080 




n. 


Sept. 6. 236691 37.876611 


1.5783596 


m. 


Oct. 14. 119260 37.882569 


1.5784395 



319 



Hence the corrected logarithms of the quantities &, &" become 9.8140209, 
and 9.8139410. 

We are now, according to the precept of Art. 146, to commence the determi- 
nation of the elements from the quantities/, /, r", 6, and to continue the calcula- 
tion so far as to obtain rj, and again from the quantities /", r, /, ^" so as to 
obtain rl'. 

logi? 0.0011576 

logrf' 0.0011552 

logP' .... 9.99^9225 

log ^ .... 9.6309476 

From the first hypothesis, therefore, there results X = 0.0000251, and 
Z=: 0.0029648. 

In the second hypothesis, we assign to P and Q the values which we find 
in the first hypothesis for P' and Q. We put, therefore, 

^ = log P= 9.9999225, 
j/ = log^= 9.6309476. 

Since the computation is to be performed in precisely the same manner as in 
the first hypothesis, it is sufficient to set down here its principal results : — 



oj 5°43'56'M0 

w + (J 7 49 1 .97 

log Qc^vciiii 0.9142633 



log / 

log-; 



r 59' 34" 98 
. 0.4749037 
. 0.7724177 



320 



APPENDIX. 



. 0.7724952 
185° 10' 39" 64 
189 25 42 .36 



logr 0.4748696 

loo-r" .• . 0.4743915 



i {u + m) 

2/' . . 
2/. . . 
2/" . . 



264° 21' 50" 
288 49 5 
.13 53 58 
. 6 57 15 
. 6 56 43 



.64 
.57 
82 
58 
41 



In this case we distribute the difference 0".17 so as to make 2/= 6° 51' 15".49 
and 2/"= 6° 56' 43".33. 

It would not be worth while to compute anew the reductions of the time on 
account of the aberration, for they scarcely differ 1" from those which we de- 
rived from the first hypothesis. 
Further computations furnish 

log 7] = 0.0011582, log?]" = 0.0011558, whence are deduced 
log P'= 9.9999225, X= 0.0000000 
log ^=9.6309955, Y= 0.0000479. 

From which it is apparent how much more exact the second hypothesis is than 
the first. 

For the sake of completing the example, we will still construct the third 
hypothesis, in which we shall adopt the values of P' and Q' derived from the 
second hypothesis for the values of P and Q. 

Putting, therefore, 

a; = log Pz= 9.9999225 
^ = \ogQ = 9.6309955 

the following are obtained for the most important parts of the computation : — 



0) 5°43'56".10 



(0 -\-a . . . 
log Q c sin 0) 



log r . 
log — 



7 49 1 .97 
0.9143111 

7° 59' 35".02 
0.4749031 

0.7724168 



. 0.7724943 
185° 10' 39".69 



t;' 189° 25' 42".45 

logr 0.4748690 

logr" 0.4743909 

^{ti'J^u) .... 264° 21' 50".64 
^(i^"_M) .... 288 49 5 .57 

2/' 13 53 58 .94 

2/ 6 57 15 .65 

2/" 6 56 43 .49 



APPENDIX. 



321 



The difference 0".2 between 2/' and 2/ -j- If" is divided as in the first 
hypothesis, making 2/= 6° 57' 15".55, and 2/"= 6° 56' 43".39. 

All these numbers differ so little from those given by the second hypothesis 
that it may safely be concluded that the third hypothesis requires no further cor- 
rection ; if the computation should be continued as in the preceding hypotheses, 
the result would be X=: 0.0000000, r= 0.0000001, which last value must be 
regarded as of no consequence, and not exceeding the unavoidable uncertainty 
belonging to the last decimal figure. 

We are, therefore, at liberty to proceed to the determination of the elements 
from 2/', r, r", ^' according to the methods contained in articles 88-97. 

The elements are found to be as follows : — 

Epoch of the mean longitude, 1845, .... 278° 47' 13".79 

Mean daily motion, .... .... 771".5855 

Longitude of the perihelion, 148° 27' 49".70 

Angle of eccentricity, 4 33 28 .35 

Logarithm of the major semi-axis .... 0.4417481 

Longitude of the ascending node, .... 80° 46' 36".94 

Inchnation of the orbit, 10 37 7 .98 

The computation of the middle place from these elements gives 

a'= 324° 34' 5r.05, ^'^ — 14° 45' 28".31 

which differ but little from the observed values 



«'=324°34'50".92, 



14°45'28".28. 



41 



322 APPENDIX. 



FORMULAS FOR COMPUTING THE ORBIT OF A COMET. 

Given 

Mean times of the observations in days, H , f, t'" 

Observed longitudes of the comet, a', a", a'" 

Observed latitudes of the comet, (i', ft", ft'" 

Longitudes of the sun, A, A', A" 

Distances of the sun from the earth, R', R", R"' 

Required 

The curtate distances from the earth, q', q", q"' 

Compute 

I. 

tan^" j^ f— f m sin (a' — A") — tan (i' 

sin (a" — A") f — t^ tan /3'" — msin(a'" — A") 

and by means of this, approximately, 

q"' = Mq\ 

n. 

R"' cos {A"— A) — Rf=zffcos{G — A) 
R!" sin {A"— A) =:.gmi{G — A) 

g is the chord of the earth's orbit between the first and third places of the earth. 
G the longitude of the first place of the earth as seen from the third place. 

III. 

M— cos {a'"— a') = hcos^ cos {R— a"') 
sin («'" — a') = k cos ^ sin {R — a'") 
if tan ft"' — tan ft' =i h sin ^. 
h is always positive. If iVis a point, the coordinates of which, referred to the 
third place of the earth, are 

q' cos a', q' sin a', q tan ft, 
then are 

hq', H, t, 



APPENDIX. 323 

the polar coordinates of the third place of the comet, (that is, the distance, longi- 
tude and latitude,) referred to the point i\^ as the origin. 

IV. 
cos C cos ( 6^ — ^) = cos 9 ^sm(p=A 

cos (^ COS {a' — A') = cos Y R' sin \^'^= B' 

cos r cos («'" — A^") = cos v^'" R'" sin ^>"' = B'" 
By means of 9 , ^i', ip'", A, B\ B"\ Olbers's formulas, become : — 
F ==(>^9'— ^cos9)2+^2 

/2 = (q' sec {^' — R' cos x^J 4- B"" 

r""" = {M^' sec {i'" — R'" cos ^>"'f -f ^'"^ 

The computation would be somewhat easier by 



V. 

cos (i'^^f, g cos 9 — /' R! cos »f '= c' 
—Kr-=f gcos^i'—f'R cost/^ =c 



in which 



u=ihq' — g cos 9 

VI. 
A value of u is to be found by trial which will satisfy the equation 

(/ + ,-+ /,)-| _(/+ r"'-Jcf = ^, 

in which 

log w/= 0.9862673 

If no approximate value for q' or for / or r" is otherwise known, by means 
of which an approximate value of u can be found, we may begin with 



324 ' APPENDIX. 

This trial will be facilitated by Table ITIa, which gives fi corresponding to 

"?— (/+/")!' 
by means of which is found k, which corresponds rigorously to r, r"', and f" — f: — 

in which 

log x = 8.5366114. 

The process may be as follows : For any value of u compute k, /, /", by V, 

and with r', /", compute r], with which fi, is to be taken from Table Ilia, and a value 

of k is to be computed which corresponds to the /, r"', f — i! used. And m is to 

be changed until the second value of k shall agree exactly with that computed 

byV. 

Then we have 

_/ M + g'cosq) 

^ — h 

vn. 

q' cos {a' — A) — R' =zr cos h' cos [t — A) 

(/ sin {a' — A) = / cos h' sin [t — A) 

q' tan {V = r sin h' 

^"' cos {a'" — A") — R" = r'" cos V" cos {t" — A") 

q'" sin {a" — A") = r" cos h'" sin {t" — A'') 

q"' tan (i'" = r" sin V". 

FIRST CONTROL. 

The values of /, /", obtained from these formulas, must agree exactly with 
those before computed. 

t, y ; I" , h'", are heliocentric longitudes and latitudes of the comet. 

The motion is direct when f — I' is positive, and retrograde when f' — t is 
negative. 



APPENDIX. 325 

vni. 

± tan V = tan i sin {J — ^ ) 

, tanJ'" — tan&'cos(r'— Z') , . ,,, ^, 
=*= sm(r-0 ^ = tanecos^/— S2) 

i the inclinatioi] is always positive, and less than 90°. The upper signs are to be 
used when the motion is direct j the lower when it is retrograde. 

IX. 

tan (Z' — S2 ) , IT, „ X tan ill" — g^) . /t-w r^\ 

— ^ — -. '- = tan (Jj — Q), — ^ r^^= tan (L — Q, ). 

cos ^ \ /' cos I ^ ' 

I! and L'" are the longitudes in orbit. 

SECOND CONTROL. 

The value of It before computed must be exactly 

^ = V^ [r'2 + r""" — 2/ r'" cos {U" — Z' )]. 

X. 

1 cos 1 {L' 7t) 



cos {II" — L') cosec \ {L'" — L') sin i {L' — n) 

\J r' \j r'" \j q 

n, the longitude of the perihelion, is counted from a point in the orbit from which 
the distance, in the direction" of the order of the signs, to the ascending node, is 
equal to the longitude of the ascending node. 

XI. 

The true anomalies are 

v'=r—7i, v"'=L'"—ii. 

With these the corresponding M' and M"' are to be taken from Barker's 
Table, and we have then the time of perihelion passage 



326 APPENDIX. 

in which M' and M'" have the sign of v and v" ; the constant log n is 
log n = 0.0398723. 

The upper signs serve for direct, the lower for retrograde motion. 
For the use of Table TLa instead of Bakker's Table, see Article 18 of the 
Appendix. 

THIRD CONTROL. 

The two values of T, from H, and f, must agree exactly. 

XII. 

With T, q, 71, Q,, i, I", A', R", compute a" and (i", and compare them with the 
observed values. And also compute with these values the formula 

tan S" 
sin {a — A ) 

If this value agrees with that of m of formulas I., the orbit is exactly deter- 
mined according to the principles of Olbers's Method. That is, while it satisfies 
exactly the two extreme places of the comet, it agrees with the observations in 
the great circle which connects the middle place of the Comet with the middle 
place of the Sun. 

If a difference is found, M can be changed until the agreement is complete. 



TABLES. 



TABLE I. (See articles 42, 45.) 





ELLIPSE. 




HYPERBOLA. 


A 


LogB 


C 


T 


LogB 


C 


T 


0.000 








0.00000 








0.00000 


.001 








.00100 










.00100 


.002 





2 


.00200 







2 


.00200 


.003 


1 


4 


.00301 




1 


4 


.00299 


.004 


1 


7 


.00401 




1 


7 


.00399 


0.005 


2 


11 


0.00502 




2 


11 


0.00498 


.006 


3 


16 


.00603 




3 


16 


.00597 


.007 


4 


22 


.00704 




4 


22 


.00696 


.008 


5 


29 


.00805 




5 


29 


.00795 


.009 


6 


37 


.00907 




6 


37 


.00894 


0.010 


7 


46 


0.01008 




7 


46 


0.00992 


.011 


9 


56 


.01110 




9 


55 


.01090 


.012 


11 


66 


.01212 




11 


66 


.01189 


.013 


13 


78 


.01314 




13 


77 


.01287 


.014 


15 


90 


.01416 




15 


89 


.01384 


0.015 


17 


103 


0.01518 




17 


102 


0.01482 


.016 


19 


118 


.01621 


1 19 


116 


.01580 


.017 


22 


133 


.01723 




21 


131 


.01677 


.018 


24 


149 


.01826 




24 


147 


.01774 


.019 


27 


166 


.01929 




27 


164 


.01872 


0.020 


30 


184 


0.02032 




30 


182 


0.01968 


.021 


33 


203 


.02136 




33 


200 


.02065 


,022 


36 


223 


.02239 




36 


220 


.02162 


.023 


40 


244 


.02343 




39 


240 


.02258 


.024 


43 


265 


.02447 




43 


261 


.02355 


0.025 


47 


288 


0.02551 




46 


283 


0.02451 


.026 


51 


312 


.02655 




50 


306 


.02547 


.027 


55 


336 


.02760 




54 


330 


.02643 


.028 


59 


362 


.02864 




58 


355 


.02739 


.029 


63 


388 


.02969 




62 


381 


.02834 


0.030 


67 


416 


0.03074 




67 


407 


0.02930 


.031 


72 


444 


.03179 




71 


435 


.03025 


.032 


77 


473 


.03284 




76 


463 


.03120 


.033 


82 


503 


.03389 




80 


492 


.03215 


.034 


87 


535 


.03495 




85 


523 


.03310 


0.035 


92 


567 


0.03601 




91 


554 


0.03404 


.036 


97 


600 


.03707 




96 


585 


.03499 


.037 


103 


634 


.03813 




101 


618 


.03593 


.038 


108 


669 


.03919 




107 


652 


.03688 


.039 


114 


704 


.04025 




112 


686 


.03782 


.040 


120 


741 


.04132 




118 


722 


.03876 



TABLE 





ELLIPSE. 




HYPERBOLA. 1 


A 


LogB 


C 


T 


LogB 


c 


T 


0.040 


120 


741 


0.041319 


118 


722 


0.038757 


.041 


126 


779 


.042387 




124 


758 


.039695 


.042 


133 


818 


.043457 




130 


795 


.040632 


.043 


139 


858 


.044528 




136 


833 


.041567 


.044 


146 


898 


.045601 




143 


872 


.042500 


0.045 


152 


940 


0.046676 




149 


912 


0.043432 


.046 


159 


982 


.047753 




156 


953 


.044363 


.047 


166 


1026 


.048831 




163 


994 


.045292 


.048 


173 


1070 


.049911 




170 


1037 


.046220 


.049 


181 


1116 


.050993 




177 


1080 


.047147 


0.050 


188 


1162 


0.052077 




184 


1124 


0.048072 


.051 


196 


1210 


.053163 




191 


1169 


.048995 


.052 


204 


1258 


.054250 




199 


1215 


.049917 


.053 


212 


1307 


.055339 




207 


1262 


.050838 


.054 


220 


1358 


.056430 




215 


1310 


.051757 


0.055 


228 


1409 


0.057523 




223 


1358 


0.052675 


.056 


286 


1461 


.058618 




231 


1407 


.053592 


.057 


245 


1514 


.059714 




239 


1458 


.054507 


.058 


254 


1568 


.060812 




247 


1509 


.055420 


.059 


263 


1623 


.061912 




25 G 


1561 


.056332 


0.060 


272 


1679 


0.063014 




265 


1614 


0.057243 


.061 


281 


1736 


.064118 




273 


1667 


.058152 


.062 


290 


1794 


.065223 




282 


1722 


.059060 


.063 


300 


1853 


.066331 




291 


1777 


.059967 


.064 


309 


1913 


.067440 




301 


1833 


.060872 


0.065 


319 


1974 


0.068551 




310 


1891 


0.061776 


.066 


329 


2036 


.069664 




320 


1949 


.062678 


.067 


339 


2099 


.070779 




329 


2007 


.063579 


.068 


350 


2163 


.071896 




339 


2067 


.064479 


.069 


360 


2228 


.073014 




349 


2128 


.065377 


0.070 


371 


2294 


0.074135 




359 


2189 


0.066274 


.071 


381 


2360 


.075257 




370 


2251 


.067170 


.072 


392 


2428 


.076381 




380 


2314 


.068064 


.073 


403 


2497 


.077507 




390 


2378 


.068957 


.074 


415 


2567 


.078635 




401 


2443 


.069848 


0.075 


426 


2638 


0.079765 




412 


2509 


0.070738 


.076 


437 


2709 


.080897 




423 


2575 


.071627 


.077 


449 


2782 


.082030 




434 


2643 


.072514 


.078 


461 


2856 


.083166 




445 


2711 


.073400 


.079 


473 


2930 


.084303 




457 


2780 


.074285 


.080 


485 


3006 


.085443 




468 


2850 


.075168 



TABLE I, 





ELLIPSE. 




HYPERBOLA. 


A 


LogB 


C 


T 


LogB 


C 


T 


0.080 


485 


3006 


0.085443 


468 


2850 


0.075168 


.081 


498 


3083 


.086584 




480 


2921 


.076050 


.082 


510 


3160 


.087727 




492 


2992 


.076930 


.083 


523 


3239 


.088872 




504 


3065 


.077810 


.084 


535 


3319 


.090019 




516 


3138 


.078688 


0.085 


548 


3399 


0.091168 




528 


3212 


0.079564 


.086 


561 


3481 


.092319 




540 


3287 


.080439 


.087 


575 


3564 


.093472 




553 


3363 


.081313 


.088 


588 


3647 


.094627 




566 


3440 


.082186 


.089 


602 


3732 


.095784 




578 


3517 


.083057 


0.090 


615 


3818 


0.096943 




591 


3595 


0.083927 


.091 


629 


3904 


.098104 




604 


3674 


.084796 


.092 


643 


3992 


.099266 




618 


3754 


.085663 


.093 


658 


4081 


.100431 




631 


3835 


.086529 


.094 


672 


4170 


.101598 




645 


3917 


.087394 


0.095 


687 


4261 


0.102766 




658 


3999 


0.088257 


.096 


701 


4353 


.103937 




672 


4083 


.089119 


.097 


716 


4446 


.105110 




686 


4167 


.089980 


.098 


731 


4539 


.106284 




700 


4252 


.090840 


.099 


746 


4634 


.107461 




714 


4338 


.091698 


0.100 


762 


4730 


0.108640 




728 


4424 


0.092555 


.101 


777 


4826 


.109820 




743 


4512 


.093410 


.102 


793 


4924 


.111003 




758 


4600 


.094265 


.103 


809 


5023 


.112188 




772 


4689 


.095118 


.104 


825 


5123 


.113375 




787 


4779 


.095969 


0.105 


841 


5224 


0.114563 




802 


4870 


0.096820 


.106 


857 


5325 


.115754 




817 


4962 


.097669 


.107 


873 


5428 


.116947 




833 


5054 


.098517 


.108 


890 


5532 


.118142 




848 


5148 


.099364 


.109 


907 


5637 


.119339 




864 


5242 


.100209 


0.110 


924 


5743 


0.120538 




880 


5337 


0.101053 


•111 


941 


5850 


.121739 




895 


5432 


.101896 


.112 


958 


5958 


.122942 




911 


5529 


.102738 


.113 


975 


6067 


.124148 




928 


5626 


.103578 


.114 


993 


6177 


.125355 




944 


5724 


.104417 


0.115 


1011 


6288 


0.126564 




960 


5823 


0.105255 


.116 


1029 


6400 


.127776 




977 


5923 


.106092 


.117 


1047 


6513 


.128989 




994 


6024 


.106927 


.118 


1065 


6627 


.130205 




1010 


6125 


.107761 


.119 


1083 


6742 


.131423 




1027 


6228 


.108594 


.120 


1102 


6858 


.132643 




1045 


G331 


.109426 

1 



TABLE 1, 





ELLIPSE. 




HYPERBOLA. 1 


A 


LogB 


C 


T 


LogB 


C 


T 


0.120 


1102 


6858 


0.132643 


1045 


6331 


0.109426 


.121 


1121 


6976 


.133865 




1062 


6435 


.110256 


.122 


1139 


7094 


.135089 




107-9 


6539 


.111085 


.123 


1158 


7213 


.136315 




1097 


6645 


.111913 


.124 


1178 


7334 


.137543 




1114 


6751 


.112740 


0.125 


1197 


7455 


0.138774 




1132 


6858 


0.113566 


.126 


1217 


7577 


.140007 




1150 


6966 


.114390 


.127 


1236 


7701 


.141241 




1168 


7075 


.115213 


.128 


1256 


7825 


.142478 




1186 


7185 


.116035 


.129 


1276 


7951 


.143717 




1205 


7295 


.116855 


0.130 


1296 


8077 


0.144959 




1223 


7406 


0.117075 


.131 


1317 


8205 


.146202 




1242 


7518 


.118493 


.132 


1337 


8334 


.147448 




1261 


7631 


.119310 


.133 


1358 


8463 


.148695 




1280 


7745 


.120126 


.134 


1378 


8594 


.149945 




1299 


7859 


.120940 


0.135 


1399 


8726 


0.151197 




1318 


7974 


0.121754 


.136 


1421 


8859 


.152452 




1337 


8090 


.122566 


.137 


1442 


8993 


.153708 




1357 


8207 


.123377 


.138 


1463 


9128 


.154967 




1376 


8325 


.124186 


.139 


1485 


9264 


.156228 




1396 


8443 


.124995 


0.140 


1507 


9401 


0.157491 




1416 


8562 


0.125802 


.141 


1529 


9539 


.158756 




1436 


8682 


.126609 


.142 


1551 


9678 


.160024 




1456 


8803 


.127414 


.143 


1573 


9819 


.161294 




1476 


8925 


.128217 


.144 


1596 


9960 


.162566 




1497 


9047 


.129020 


0.145 


1618 


10102 


0.163840 




1517 


9170 


0.129822 


.146 


1641 


10246 


.165116 




1538 


9294 


.130622 


.147 


1664 


10390 


.166395 




1559 


9419 


.131421 


.148 


1687 


10536 


.167676 




1580 


9545 


.132219 


.149 


1710 


10683 


.168959 




1601 


9671 


.133016 


0.150 


1734 


10830 


0.170245 




1622 


9798 


0.133812 


.151 


1757 


10979 


.171533 




1643 


9926 


.134606 


.152 


1781 


11129 


.172823 




1665 


10055 


.135399 


.153 


1805 


11280 


.174115 




1686 


10185 


.136191 


.154 


1829 


11432 


.175410 




1708 


10315 


.136982 


0.155 


1854 


11585 


0.176707 




1730 


10446 


0.137772 


.156 


1878 


11739 


.178006 




1752 


10578 


.138561 


.157 


1903 


11894 


.179308 




1774 


10711 


.139349 


.158 


1927 


12051 


.180612 




1797 


10844 


.140135 


.159 


1952 


12208 


.181918 




1819 


10978 


.140920 


.160 


1977 


12366 


.183226 




1842 


11113 


.141704 



TABLE I. 





ELLIPSE. 




HYPEEBOLA. 


A 


LogB 


C 


T 




LogB 


C 


T 


0.160 


1977 


12366 


0.183226 


1842 


11113 


0.141704 


.161 


2003 


12526 


.184537 




1864 


11249 


.142487 


.162 


2028 


12686 


.185850 




1887 


11386 


.143269 


.163 


2054 


12848 


.187166 




1910 


11523 


.144050 


.164 


2080 


13011 


.188484 




1933 


11661 


.144829 


0.165 


2106 


13175 


0.189804 




1956 


11800 


0.145608 


.166 


2132 


13340 


.191127 




1980 


11940 


.146385 


.167 


2158 


13506 


.192452 




2003 


12081 


.147161 


.168 


2184 


13673 


.193779 




2027 


12222 


.147937 


.169 


2211 


13841 


.195109 




2051 


12364 


.148710 


0.170 


2238 


14010 


0.196441 




2075 


12507 


0.149483 


.171 


2265 


14181 


.197775 




2099 


12651 


.150255 


.172 


2292 


14352 


.199112 




2123 


12795 


.151026 


.173 


2319 


14525 


.200451 




2147 


12940 


.151795 


.174 


2347 


14699 


.201793 




2172 


13086 


.152564 , 


0.175 


2374 


14873 


0.203137 




2196 


13233 


0.153331 


.176 


2402 


15049 


.204484 




2221 


13380 


.154097 


.177 


2430 


15226 


.205832 




2246 


13529 


.154862 


.178 


2458 


15404 


.207184 




2271 


13678 


.155626 


.179 


2486 


15583 


.208538 




2296 


13827 


.156389 


0.180 


2515 


15764 


0.209894 




2321 


13978 


0.157151 


.181 


2543 


15945 


.211253 




2346 


14129 


.157911 


.182 


2572 


16128 


.212614 




2372 


14281 


.158671 


.183 


2601 


16311 


.213977 




2398 


14434 


.159429 


.184 


2630 


16496 


.215343 




2423 


14588 


.160187 


0.185 


2660 


16682 


0.216712 




2449 


14742 


0.160943 


.186 


2689 


16868 


.218083 




2475 


14898 


.161698 


.187 


2719 


17057 


.219456 




2502 


15054 


.162453 


.188 


2749 


17246 


.220832 




2528 


15210 


.163206 


.189 


2779 


17436 


.222211 




2554 


15368 


.163958 


0.190 


2809 


17627 


0.223592 




2581 


15526 


0.164709 


.191 


2839 


17820 


.224975 




2608 


15685 


.165458 


.192 


2870 


18013 


.226361 




2634 


15845 


.166207 


.193 


2900 


18208 


.227750 




. 2661 


16005 


.166955 


.194 


2931 


18404 


.229141 




2688 


16167 


.167702 


0.195 


2962 


18601 


0.230535 




2716 


16329 


0.168447 


.196 


2993 


18799 


.231931 




2743 


16491 


.169192 


.197 


3025 


18998 


.233329 




2771 


16655 


.169935 


.198 


3056 


19198 


.234731 




2798 


16819 


.170678 


.199 


3088 


19400 


.236135 




2826 


16984 


.171419 


.200 


3120 


19602 


.237541 




2854 


17150 


.172159 



TABLE I. 





ELLIPSE. 




HYPERBOLA. 


A 


LogB 


C 


T 


LogB 


C 


T 


0.200 


3120 


19602 


0.237541 


2854 


17150 ( 


).172159 


.201 


3152 


19806 


.238950 




2882 


17317 


.172899 


.202 


3184 


200U 


.240361 




2910 


17484 


.173637 


.203 


3216 


20217 


.241776 




2938 


17652 


.174374 


.204 


3249 


20424 


.243192 




2967 


17821 


.175110 


0.205 


3282 


20632 


0.244612 




2995 


17991 ( 


).l 75845 


.206 


3315 


20842 


.246034 




3024 


181G1 


.176579 


.207 


3348 


21052 


.247458 




3053 


18332 


.177312 


.208 


3381 


21264 


.248885 




3082 


18504 


.178044 


.209 


3414 


21477 


.250315 




3111 


18677 


.178775 


0.210 


3448 


21690 


0.251748 




3140 


18850 ( 


).179505 


.211 


3482 


21905 


.253183 




31G9 


19024 


.180234 


.212 


3516 


22122 


.254620 




3199 


19199 


.180962 


.213 


3550 


22339 


.256061 




3228 


19375 


.181688 


.214 


3584 


22557 


.257504 




3258 


19551 


.182414 


0.215 


3618 


22777 


0.258950 




3288 


19728 ( 


).183139 


.216 


3653 


22998 


.260398 




3318 


19906 


.183863 


.21,7 


3688 


23220 


.261849 




3348 


20084 


.184585 


.218 


3723 


23443 


.263303 




3378 


20264 


.185307 


.219 


3758 


23667 


.264759 




3409 


20444 


.186028 


0.220 


3793 


23892 


0.266218 




3439 


20625 ( 


).186747 


.221 


3829 


24119 


.267680 




3470 


20806 


.187466 


.222 


3865 


24347 


.269145 




3500 


20988 


.188184 


.223 


3900 


24576 


.270612 




3531 


21172 


.188900 


.224 


3936 


24806 


.272082 




3562 


21355 


.189616 


0.225 


3973 


25037 


0.273555 




3594 


21540 ( 


).l 90331 


.226 


4009 


25269 


.275031 




3625 


21725 


.191044 


.227 


4046 


25502 


.276509 




3656 


21911 


.191757 


.228 


4082 


25737 


.277990 




3688 


22098 


.192468 


.229 


4119 


25973 


.279474 




3719 


22285 


.193179 


0.230 


4156 


26210 


0.280960 




3751 


22473 ( 


).l 93889 


.231 


4194 


26448 


.282450 




3783 


22662 


.194597 


.232 


4231 


26687 


.283942 




3815 


22852 


.195305 


.233 


4269 


26928 


.285437 




3847 


23042 


.196012 


.234 


4306 


27169 


.286935 




3880 


23234 


.196717 


0.235 


4344 


27412 


0.288435 




3912 


23425 ( 


).197422 


.236 


4382 


27656 


.289939 




3945 


23618 


.198126 


.237 


4421 


27901 


.291445 




3977 


23811 


.198829 


.238 


4459 


28148 ' 


.292954 




4010 


24005 


.199530 


.239 


4498 


28395 


.294466 




4043 


24200 


.200231 


.210 


4537 


28644 


.295980 




4076 


24396 


.200931 



TABLE I, 





ELLIPSE. 




HYPERBOLA. 


A 


LogB 


C 


T 


LogB 


C 


T 


0.240 


4537 


28644 


0.295980 


4076 


24396 


0.200931 


.241 


4576 


28894 


.297498 




4110 


24592 


.201630 


.242 


4615 


29145 


.299018 




4143 


24789 


.202328 


.243 


4654 


29397 


.300542 




4176 


24987 


.203025 


.244 


4694 


29651 


.302068 




4210 


25185 


.203721 


0.245 


4734 


29905 


0.303597 




4244 


25384 


0.204416 


.246 


4774 


30161 


.305129 




4277 


25584 


.205110 


.247 


4814 


30418 


.306664 




4311 


25785 


.205803 


.248 


4854 


30676 


.308202 




4346 


25986 


.206495 


.249 


4894 


30935 


.309743 




4380 


26188 


.207186 


0.250 


4935 


31196 


0.311286 




4414 


26391 


0.207876 


.251 


4976 


31458 


.312833 




4449 


26594 


.208565 


.252 


5017 


31721 


.314382 




4483 


26799 


.209254 


.253 


5058 


31985 


.315935 




4518 


27004 


.209941 


.254 


5099 


32250 


.317490 




4553 


27209 


.210627 


0.255 


5141 


32517 


0.319048 




4588 


27416 


0.211313 


.256 


5182 


32784 


.320610 




4623 


27623 


.211997 


.257 


5224 


33053 


.322174 




4658 


27830 


.212681 


.258 


5266 


33323 


.323741 




4694 


28039 


.213364 


.259 


5309 


33595 


.325312 


* 


4729 


28248 


.214045 


0.260 


5351 


33867 


0.326885 




4765 


28458 


0.214726 


.261 


5394 


34141 


,328461 




4801 


28669 


.215406 


.262 


5436 


34416 


.330041 




4838 


28880 


.216085 


.263 


5479 


34692 


.331623 




4873 


29092 


.216763 


.264 


5522 


34970 


.333208 




4909 


29305 


.217440 


0.265 


5566 


35248 


0.334797 




4945 


29519 


0.218116 


.266 


5609 


35528 


.336388 




4981 


29733 


.218791 


.267 


5653 


35809 


.337983 




5018 


29948 


.219465 


.268 


5697 


36091 


.339580 




5055 


30164 


.220138 


.269 


5741 


36375 


.341181 




5091 


30380 


.220811 


0.270 


5785 


36659 


0.342785 




5128 


30597 


0.221482 


.271 


5829 


36945 


.344392 




5165 


30815 


.222153 


.272 


5874 


37232 


.346002 




5202 


31033 


.222822 


.273 


5919 


37521 


.347615 




5240 


31253 


.223491 


.274 


5964 


37810 


.349231 




5277 


31473 


.224159 


0.275 


6009 


38101 


0.350850 




5315 


31693 


0.224826 


.276 


6054 


38393 


.352473 




5352 


31915 


.225492 


.277 


6100 


38686 


.354098 




5390 


32137 


.226157 


.278 


6145 


38981 


.355727 




5428 


32359 


.226821 


.279 


6191 


39277 


.357359 




5466 


32583 


.227484 


.280 


6237 


39573 


.358994 




5504 


32807 


.228147 



TABLE I, 





ELLIPSE. 




HYPERBOLA. 


A 


LogB 


c 


T 


LogB 
5504 


C 


T 


0.280 


G237 


39573 


0.358994 


32807 


0.228147 


.281 


6283 


39872 


.360632 




5542 


33032 


.228808 1 


.282 


6330 


40171 


.362274 




5581 


33257 


.229469 


.283 


6376 


40472 


.363918 




5619 


33484 


.230128 ! 


.284 


6423 


40774 


.365566 




5658 


33711 


.230787 


0.285 


6470 


41077 


0.367217 




5697 


33938 


0.231445 


.286 


6517 


41381 


.368871 




5736 


34167 


.232102 


.287 


6564 


41687 


.370529 




5775 


34396 


.232758 


.288 


6612 


41994 


.372189 




5814 


34626 


.233413 


.289 


6660 


42302 


.373853 




5853 


34856 


.234068 


0.290 


6708 


42611 


0.375521 




5893 


35087 


0.234721 


.291 


6756 


42922 


.377191 




5032 


35319 


.235374 


.292 


6804 


43233 


. .378865 




5972 


35552 


.236025 


.293 


6852 


43547 


.380542 




6012 


35785 


.236676 


.294 


6901 


43861 


.382222 




6052 


36019 


.237326 


0.295 


6950 


44177 


0.383906 




6092 


36253 


0.237975 


.296 


6999 


44493 


.385593 




6132 


36489 


.238623 


-.297 


7048 


44812 


.387283 




6172 


36725 


.239271 


.298 


7097 


45131 


.388977 




6213 


36961 


.239917 


.299 


7147 


45452 


.390673 




6253 


37199 


.240563 


j .300 


7196 


45774 


.392374 




6294 


37437 


.241207 



TABLE 11. (See Article 93.) 



h 


logyy 


h 


logyy 


h 


logyy 


0.0000 


0.0000000 


0.0040 


0.0038332 


0.0080 


0.0076133 


.0001 


.0000965 


.0041 


.0039284 


.0081 


.0077071 


.0002 


.0001930 


.0042 


.0040235 


.0082 


.0078009 


.0003 


.0002894 


.0043 


.0041186 


.0083 


.0078947 


.0004 


.0003858 


.0044 


.0042136 


.0084 


.0079884 


0.0005 


0.0004821 


0.0045 


0.0043086 


0.0085 


0.0080821 


.0006 


.0005784 


.0046 


.0044036 


.0086 


.0081758 


.0007 


.0006747 


.0047 


.0044985 


.0087 


.0082694 


.0008 


.0007710 


.0048 


.0045934 


.0088 


.0083630 


.0009 


.0008672 


.0049 


.0046883 


.0089 


.0084566 


0.0010 


0.0009634 


0.0050 


0.0047832 


0.0090 


0.0085502 


, .0011 


.0010595 


.0051 


.0048780 


.0091 


.0086437 


.0012 


.0011556 


.0052 


.0049728 


.0092 


.0087372 


.0013 


, .0012517 


.0053 


.0050675 


.0093 


.0088306 


.0014 


.0013478 


.0054 


.0051622 


.0094 


.0089240 


0.0015 


0.0014438 


0.0055 


0.0052569 


0.0095 


0.0090174 


.0016 


.0015398 


.0056 


.0053515 


.0096 


.0091108 


.0017 


.0016357 


.0057 


.0054462 


.0097 


.0092041 


.0018 


.0017316 


.0058 


.0055407 


.0098 


.0092974 


.!.)19 


.0018275 


.0059 


.0056353 


.0099 


.0093906 


0.0020 


0.0019234 


0.0060 


0.0057298 


0.0100 


0.0094838 


.0021 


.0020192 


.0061 


.0058243 


.0101 


.0095770 


.0022 


.0021150 


.0062 


.0059187 


.0102 


.0096702 


.0023 


.0022107 


.0063 


.0060131 


.0103 


.0097633 


.0024 


.0023064 


.0064 


.0061075 


.0104 


.0098564 


0.0025 


0.0024021 


0.0065 


0.0062019 


0.0105 


0.0099495 


.0026 


.0024977 


.0066 


.0062962 


.0106 


.0100425 


.0027 


.0025933 


.0067 


.0063905 


.0107 


.0101355 


.0028 


.0026889 


.0068 


.0064847 


.0108 


.0102285 


.0029 


.0027845 


.0069 


.0065790 


.0109 


.0103215 


0.0030 


0.0028800 


0.0070 


0.0066732 


0.0110 


0.0104144 


.0031 


.0029755 


.0071 


.0067673 


.0111 


.0105073 


.0032 


.0030709 


.0072 


.0068614 


.0112 


.0106001 


.0033 


.0031663 


.0073 


.0069555 


.0113 


.0106929 


.0034 


.0032617 


.0074 


.0070496 


.0114 


.0107857 1 


0.0035 


0.0033570 


0.0075 


0.0071436 


0.0115 


i 
0.0108785 1 


.0036 


.0034523 


.0076 


.0072376 


.0116 


.0109712 


.0037 


.0035476 


.0077 


.0073316 


.0117 


.0110639 


.0038 


.0036428 


.0078 


.0074255 


.0118 


.0111565 


.0039 


.0037380 


.0079 


.0075194 


.0119 


.0112491 


.0040 


.0038332 


.0080 


.0076133 


.0120 


.0113417 



10 



TABLE II. 



h 


logyy 


h 


logyy 


h 


logyy 


0.0120 


0.0113417 


0.0160 


0.0150202 


0.0200 


0.0186501 


.0121 


.0114343 


.0161 


.0151115 


.0201 


.0187403 


.0122 


.0115268 


.0162 


.0152028 


.0202 


.0188304 


.0123 


.0116193 


.0163 


.0152941 


.0203 


.0189205 


.0124 


.0117118 


.0164 


.0153854 


.0204 


.0190105 


0.0125 


0.0118043 


0.0165 


0.0154766 


0.0205 


0.0191005 


.0126 


.0118967 


.0166 


.0155678 


.0206 


.0191905 


.0127 


.0119890 


.0167 


.0156589 


.0207 


.0192805 


.0128 


.0120814 


.0168 


.0157500 


.0208 


.0193704 


.0129 


.0121737 


.0169 


.0158411 


.0209 


.0194603 


0.0130 


0.0122660 


0.0170 


0.0159322 


0.0210 


0.0195502 


.0131 


.0123582 


.0171 


.0160232 


.0211 


.0196401 


.0132 


.0124505 


.0172 


.0161142 


.0212 


.0197299 


.0133 


.0125427 


.0173 


.0162052 


.0213 


.0198197 


.0134 


.0126348 


.0174 


.0162961 


.0214 


.0199094 


0.0135 


0.0127269 


0.0175 


0.0163870 


0.0215 


0.0199992 


.0136 


.0128190 


.0176 


.0164779 


.0216 


.0200889 


.0137 


.0129111 


.0177 


.0165688 


.0217 


.0201785 


.0138 


.0130032 


.0178 


.0166596 


.0218 


.0202682 


.0139 


.0130952 


.0179 


.0167504 


.0219 


.020357.S 


0.0140 


0.0131871 


0.0180 


0.0168412 


0.0220 


0.0204474 


.0141 


.0132791 


.0181 


.0169319 


.0221 


.0205369 


.0142 


.0133710 


.0182 


.0170226 


.0222 


.0206264 


.0143 


.0134629 


.0183 


.0171133 


.0223 


.0207159 


.0144 


.0135547 


.0184 


.0172039 


.0224 


.0208054 


0.0145 


0.0136465 


0.0185 


0.0172945 


0.0225 


0.0208948 


.0146 


.0137383 


.0186 


.0173851 


.0226 


.0209842 


.0147 


.0138301 


.0187 


.0174757 


.0227 


.0210736 


.0148 


.0139218 


.0188 


.0175662 


.0228 


.0211630 


.0149 


.0140135 


.0189 


.0176567 


.0229 


.0212523 


0.01,50 


0.0141052 


0.0190 


0.0177471 


0.0230 


0.0213416 


.0151 


.0141968 


.0191 


.0178376 


.0231 


.0214309 


.0152 


.0142884 


.0192 


.0179280 


.0232 


.0215201 


.0153 


.0143800 


.0193 


.0180183 


.0233 


• .0216093 


.0154 


.0144716 


.0194 


.0181087 


.0234 


.0216985 


0.0155 


0.0145631 


0.0195 


0.0181990 


0.0235 


0.0217876 


.0156 


.0146546 


.0196 


.0182893 


.0236 


.0218768 


.0157 


.0147460 


.0197 


.0183796 


.02;!7 


.0219659 


.0158 


.0148374 


.0198 


.0184698 


.0238 


.0220549 


.0159 


.0149288 


.0199 


.0185600 


.0239 


.0221440 


.0160 


.0150202 


.0200 


.0186501 


.0240 


.0222330 

1 



TABLE II. 



11 





h 


logyy 


h 


logyy 


h 


logyy 




0.0240 


0.0222330 


0.0280 


0.0257700 


0.0320 


0.0292626 




.0241 


.0223220 


. .0281 


.0258579 


.0321 


.0293494 




.0242 


.0224109 


.0282 


.0259457 


.0322 


.0294361 




.0243 


.0224998 


.0283 


.0260335 


.0323 


.0295228 




.0244 


.0225887 


.0284 


.0261213 


.0324 


.0296095 




0.0245 


0.0226776 


0.0285 


0.0262090 


0.0325 


0.0296961 




.0246 


.0227664 


.0286 


.0262967 


.0326 


.0297827 




.0247 


.0228552 


.0287 


.0263844 


.0327 


.0298693 




.0248 


.0229440 


.0288 


.0264721 


.0328 


.0299559 




.0249 


.0230328 


.0289 


.0265597 


.0329 


.0300424 




0.0250 


0.0231215 


0.0290 


0.0266473 


0.0330 


0.0301290 




.0251 


.0232102 


.0291 


.0267349 


.0331 


.0302154 




.0252 


.0232988 


.0292 


.0268224 


.0332 


.0303019 




.0253 


.0233875 


.0293 


.0269099 


.0333 


.0303883 




.0254 


.0234761 


.0294 


.0269974 


.0334 


.0304747 




0.0255 


0.0235647 


0.0295 


0.0270849 


0.0335 


0.0305611 




.0256 


.0236532 


.0296 


.0271723 


.0336 


.0306475 




.0257 


.0237417 


.0297 


.0272597 


.0337 


.0307338 




.0258 


.0238302 


.0298 


.0273471 


.0338 


.0308201 




.0259 


.0239187 


.0299 


.0274345 


.0339 


.0309064 




0.0260 


0.0240071 


0.0300 


0.0275218 


0.0340 


0.0309926 




.0261 


.0240956 


.0301 


.0276091 


.0341 


.0310788 




.0262 


.0241839 


.0302 


.0276964 


.0342 


.0311650 




.0263 


.0242723 


.0303 


.0277836 


.0343 


.0312512 




.0264 


.0243606 


.0304 


.0278708 


.0344 


.0313373 




0.0265 


0.0244489 


0.0305 


0.0279580 


0.0345 


0.0314234 




.0266 


.0245372 


.0306 


.0280452 


.0346 


.0315095 




.0267 


.0246254 


.0307 


.0281323 


.0347 


.0315956 




.0268 


.0247136 


.0308 


.0282194 


.0348 


.(1316816 




.0269 


.0248018 


.0309 


.0283065 


.0349 


.0317676 




0.0270 


0.0248900 


0.0310 


0.0283936 


0.0350 


0.0318536 




.0271 


.0249781 


.0311 


.0284806 


.0351 


.0319396 




.0272 


.0250662 


.0312 


.0285676 


.0352 


.0320255 




.0273 


.0251543 


.0313 


.0286546 


.0353 


.0321114 




.0274 


.0252423 . 


.0314 


.0287415 


.0354 


.0321973 




0.0275 


0.0253303 


0.0315 


0.0288284 


0.0355 


0.0322831 




.0276 


.0254183 


.0316 


.0289153 


.0356 


.0323689 




.0277 


.0255063 


.0317 


.0290022 


.0357 


.0324547 




.0278 


.0255942 


.0318 


.0290890 


.0358 


.0325405 




.0279 


.0256821 


.0319 


.0291758 


.0359 


.0326262 




.0280 


.0257700 


.0320 


.0292626 


.0360 


.0327120 



12 



TABLE II. 



h 


logyy 


h 


logyy 


h 


logyy 


0.0360 


3.0327120 


0.040 


0.0361192 


0.080 


0.0681057 


.0361 


.0327976 


.041 


.0369646 


.081 


.0688612 


.0362 


.0328833 


.042 


.0378075 


.082 


.0696146 


.0363 


.0329689 


.043 


.0386478 


.083 


.0703661 


.0304 


.0330546 


.044 


.0394856 


.084 


.0711157 


0.0365 ( 


).0331401 


0.045 


0.0403209 


0.085 


0.0718633 


.0366 


.0332257 


.046 


.0411537 


.086 


.0726090 


.0367 


.0333112 


.047 


.0419841 


.087 


.0733527 


.0368 


.0333967 


.048 


.0428121 


.088 


.0740945 


.0369 


.0334822 


.049 


.0436376 


.089 


.0748345 


0.0370 ( 


).0335677 


0.050 


0.0444607 


0.090 


0.0755725 


.0371 


.0336531 


.051 


.0452814 


.091 


.0763087 


.0372 


.0337385 


.052 


.0460997 


.092 


.0770430 


.0373 


.0338239 


.053 


.0469157 


.093 


.0777754 


.0374 


.0339092 


.054 


.0477294 


.094 


.0785060 


0.0375 ( 


).0339946 


0.055 


0.0485407 


0.095 


0.0792348 


.0376 


.0340799 


.056 


.0493496 


.096 


.0799617 


.0377 


.0341651 


.057 


.0501563 


.097 


.0806868 


.0378 


.0342504 


.058 


.0509607 


.098 


.0814101 


.0379 


.0343356 


.059 


.0517628 


.099 


.0821316 


0.0380 ( 


).0344208 


0.060 


0.0525626 


0.100 


0.0828513 


.0381 


.0345059 


.061 


.0533602 


.101 


.0835693 


.0382 


.0345911 


.062 


.0541556 


.102 


.0842854 


.0383 


.0346762 


.063 


.0549488 


.103 


.0849999 


.0384 


.0347613 


.064 


.0557397 


.104 


.0857125 


0.0385 ( 


).0348464 


0.065 


0.0565285 


0.105 


0.0864235 


.0386 


.0349314 


.066 


.0573150 


.106 


.0871327 


.0387 


.0350164 


.067 


.0580994 


.107 


.0878401 


.0388 


.0351014 


.068 


.0588817 


.108 


.0885459 


.0389 


.0351864 


.069 


.0596618 


.109 


.0892500 


0.0390 ( 


).0352713 


0.070 


0.0604398 


0.110 


0.0899523 


.0391 


.0353562 


.071 


.0612157 


.111 


.0906530 


.0392 


.0354411 


.072 


.0619895 


.112 


.091.3520 


.0393 


.0355259 


.073 


.0627612 


.113 


.0920494 


.0394 


.0356108 


.074 


.0635308 


.114 


.0927451 


0.0395 ( 


).0356956 


0.075 


0.0642984 


0.115 


0.0934391 


.0396 


.0357804 


.076 


.0650639 


.116 


.0941315 


.0397 


.0358651 


.077 


.0658274 


.117 


.0948223 


.0398 


.0359499 


.078 


.0665888 


.118 


.0955114 


.0399 


.0360346 


.079 


.0673483 


.119 


.0961990 


.0400 


.0361192 


.080 


.0681057 


.120 


.0968849 



TABLE II. 



h 


logyy 


h 


logyy 


h 


logyy 


0.120 


0.0968849 


0.160 


0.1230927 


0.200 


0.1471869 


.121 


.0975692 


.161 


.1237192 


.201 


.1477653 


.122 


.0982520 


.162 


.1243444 


.202 


.1483427 


.123 


.0989331 


.163 


.1249682 


.203 


.1489189 


.124 


.0996127 


.164 


.1255908 


.204 


.1494940 


0.125 


0.1002907 


0.165 


0.1262121 


0.205 


0.1500681 


.126 


.1009672 


.166 


.1268321 


.206 


.1506411 


.127 


.1016421 


.167 


.1274508 


.207 


.1512130 


.128 


.1023154 


.168 


.1280683 


.208 


.1517838 


.129 


.1029873 


.169 


.1286845 


.209 


.1523535 


0.130 


0.1036576 


0.170 


0.1292994 


0.210 


0.1529222 


.131 


.1043264 


.171 


.1299131 


.211 


.1534899 


.132 


.1049936 


.172 


.1305255 


.212 


.1540565 


.133 


.1056594 


.173 


.1311367 


.213 


.1546220 


.134 


.1063237 


.174 


.1317466 


.214 


.1551865 


0.135 


0.1069865 


0.175 


0.1323553 


0.215 


0.1557499 


.136 


.1076478 


.176 


.1329628 


.216 


.1563123 


.137 


.1083076 


.177 


.1335690 


.217 


.1568737 


.138 


.1089660 


.178 


.1341740 


.218 


.1574340 


.139 


.1096229 


.179 


.1347778 


.219 


.1579933 


0.140 


0.1102783 


0.180 


0.1353804 


0.220 


0.1585516 


.141 


.1109323 


.181 


.1359818 


.221 


.1591089 


.142 


.1115849 


.182 


.1365821 


.222 


.1596652 


.143 


.1122360 


.183 


.1371811 


.223 


.1602204 


.144 


.1128857 


.184 


.1377789 


.224 


.1607747 


0.145 


0.1135340 


0.185 


0.1383755 


0.225 


0.1613279 


.146 


.1141809 


.186 


.1389710 


.226 


.1618802 


.147 


.1148264 


.187 


.1395653 


.227 


.1624315 


.148 


.1154704 


.188 


.1401585 


.228 


.1629817 


.149 


.1161131 


.189 


.1407504 


.229 


.1635310 


0.150 


0.1167544 


0.190 


0.1413412 


0.230 


0.1640793 


.151 


.1173943 


.191 


.1419309 


.231 


.1646267 


.152 


.1180329 


.192 


.1425194 


.232 


.1651730 


.153 


.1186701 


.193 


.1431068 


.233 


.1657184 


.154 


.1193059 


.194 


.1436931 


.234 


.1662628 


0.155 


0.1199404 


0.195 


0.1442782 


0.235 


0.1668063 


.156 


.1205735 


.196 


.1448622 


.236 


.1673488 


.157 


.1212053 


.197 


.1454450 


.237 


.1678903 


.158 


.1218357 


.198 


.1460268 


.238 


.1684309 


.159 


.1224649 


.199 


.1466074 


.239 


.1689705 


.160 


.1230927 


.200 


.1471869 


.240 


.1695092 



14 



TABLE II. 



h 


logyy 


h 


logyy 


h 


logyy 


0.240 


0.1695092 


0.280 


0.1903220 


0.320 


0.2098315 


.241 


.1700470 


.281 


.1908249 


.321 


.2103040 


.242 


.1705838 


.282 


.1913269 


.322 


.2107759 


.243 


.1711197 


.283 


.1918281 


.323 


.2112470 


.244 


.1716547 


.284 


.1923286 


.324 


.2117174 


0.245 


0.1721887 


0.285 


0.1928282 


0.325 


0.2121871 


.246 


.1727218 


.286 


.1933271 


.326 


.2126562 


.247 


.1732540 


.287 


.1938251 


.327 


.2131245 


.248 


.1737853 


.288 


.1943224 


.328 


.2135921 


.249 


.1743156 


.289 


.1948188 


.329 


.2140591 


0.250 


0.1748451 


0.290 


0.1953145 


0.330 


0.2145253 


.251 


.1753736 


.291 


.1958094 


.331 


.2149909 


.252 


.1759013 


.292 


.1963035 


.332 


.2154558 


.253 


.1764280 


.293 


.1967968 


.333 


.2159200 


.254 


.1769538 


.294 


.1972894 


.334 


.2163835 


0.255 


0.1774788 


0.295 


0.1977811 


0.335 


0.2168464 


.256 


.1780029 


.296 


.1982721 


..336 


.2173085 


.257 


.1785261 


.297 


.1987624 


.337 


.2177700 


.258 


.1790484 


.298 


.1992518 


.338 


.2182308 


.259 


.1795698 


.299 


.1997406 


.339 


.2186910 


0.260 


0.1800903 


0.300 


0.2002285 


0.340 


0.2191505 


.261 


.1806100 


.301 


.2007157 


.341 


.2196093 


.262 


.1811288 


.302 


.2012021 


.342 


.2200675 


.263 


.1816467 


.303 


.2016878 


.343 


.2205250 


.264 


.1821638 


.304 


.2021727 


.344 


.2209818 


0.265 


0.1826800 


0.305 


0.2026569 


0.345 


0.2214380 


.266 


.1831953 


.306 


.2031403 


.346 


.2218935 


.267 


.1837098 


.307 


.2036230 


.347 


.2223483 


.268 


.1842235 


.308 


.2041050 


.348 


.2228025 


.269 


.1847363 


.309 


.2045862 


.349 


.2232561 


0.270 


0.1852483 


0.310 


0.2050667 


0.350 


0.2237090 


.271 


.1857594 


.311 


.2055464 


.351 


.2241613 


.272 


.1862696 


.312 


.2060254 


.352 


.2246130 


.273 


.1867791 


.313 


.2065037 


.353 


.2250640 


.274 


.1872877 


.314 


.2069813 


.354 


.2255143 


0.275 


0.1877955 


0.315 


0.2074581 


0.355 


0.2259640 


.276 


.1883024 


.316 


.2079342 


.356 


.2264131 


.277 


.1888085 


.317 


.2084096 


.357 


.2268615 


.278 


.1893138 


.318 


.2088843 


.358 


.2273093 


.279 


.1898183 


.319 


.2093582 


.359 


.2277565 


.280 


.1903220 


.320 


.2098315 


.360 


.2282031 



TABLE II. 



15 



h 


logyy 


h 


logyy 


h 


logyy 


0.360 


0.2282031 


0.400 


0.2455716 


0.440 


0.2620486 


.361 


.2286490 


.401 


.2459940 


.441 


.2624499 


.862 


.2290943 


.402 


.2464158 


.442 


.2628507 


.363 


.2295390 


.403 


.2468371 


.443 


.2632511 


.364 


.2299831 


.404 


.2472578 


.444 


.2636509 


0.365 


0.2304265 


0.405 


0.2476779 


0.445 


0.2640503 


.366 


.2308694 


.406 


.2480975 


.446 


.2644492 


.367 


.2313116 


.407 


.2485166 


.447 


.2648475 


.368 


.2317532 


.408 


.2489351 


.448 


.2652454 


.369 


.2321942 


.409 


.2493531 


.449 


.2656428 


0.370 


0.2326346 


0.410 


0.2497705 


0.450 


0.2660397 


.371 


.2330743 


.411 


.2501874 


.451 


.2664362 


.372 ■ 


.2335135 


.412 


.2506038 


.452 


.2668321 


.373 


.2339521 


.413 


.2510196 


.453 


.2672276 


.374 


.2343900 


.414 


.2514349 


.454 


.2676226 


0.375 


0.2348274 


0.415 


0.2518496 


0.455 


0.2680171 


.376 


.2352642 


.416 


.2522638 


.456 


.2684111 


.377 


.2357003 


.417 


.2526775 


.457 


.2688046 


.378 


.2361359 


.418 


.2530906 


.458 


.2691977 


.379 


.2365709 


.419 


.2535032 


.459 


.2695903 


0.380 


0.2370053 


0.420 


0.2539153 


0.460 


0.2699824 


.381 


.2374391 


.421 


.2543269 


.461 


.2703741 


.382 


.2378723 


.422 


.2547379 


.462 


.2707652 


.383 


.2383050 


.423 


.2551485 


.463 


.2711559 


.384 


.2387370 


.424 


.2555584 


.464 


.2715462 


0.385 


0.2391685 


0.425 


0.2559679 


0.465 


0.2719360 


.386 


.2395993 


.426 


.2563769 


.466 


.2723253 


.387 


.2400296 


.427 


.2567853 


.467 


.2727141 


.388 


.2404594 


.428 


.2571932 


.468 


.2731025 


.389 


.2408885 


.429 


.2576006 


.469 


.2734904 


0.390 


0.2413171 


0.430 


0.2580075 


0.470 


0.2738778 


.391 


.2417451 


.431 


.2584139 


.471 


.2742648 


.392 


.2421725 


.432 


.2588198 


.472 


.2746513 


.393 


.2425994 


.433 


.2592252 


.473 


.2750374 


.394 


.2430257 


.434 


.2596300 


.474 


.2754230 


0.395 


0.2434514 


0.435 


0.2600344 


0.475 


0.2758082 


.396 


.2438766 


.436 


.2604382 


.476 


.2761929 


.397 


.2443012 


.437 


.2608415 


.477 


.2765771 


.398 


.2447252 


.438 


.2612444 


.478 


.2769609 


.399 


.2451487 


.439 


.2616467 


.479 


.2773443 


.400 


.2455716 


.440 


.2620486 


.480 


.2777272 



16 



TABLE II. 



h. 


logyy 


h 


logyy 


h 


logyy 


0.480 


0.2777272 


0.520 


0.2926864 


0.560 


0.3069938 


.481 


.2781096 


.521 


.2930518 


.561 


.3073437 


.482 


.2784916 


.522 


.2934168 


.562 


.3076931 


.483 


.2788732 


.523 


.2937813 • 


.563 


.3080422 


.484 


.2792543 


.'524 


.2941455 


.564 


.3083910 


0.485 


0.2796349 


0.525 


0.2945092 


0.565 


0.3087394 


.486 


.2800151 


.526 


.2948726 


.566 


.3090874 


.487 


.2803949 


.527 


.2952355 


.567 


.3094350 


.488 


.2807743 


.528 


.2955981 


.568 


.3097823 


.489 


.2811532 


.529 


.2959602 


.569 


.3101292 


0.490 


0.2815316 


0.530 


0.2963220 


0.570 


0.3104758 


.491 


.2819096 


.531 


.2966833 


.571 


.3108220 


.492 


.2822872 


.532 


.2970443 


.572 


.3111678 


.493 


.2826644 


.533 


.2974049 


.573 


.3115133 


.494 


.2830411 


.534 


.2977650 


.574 


.3118584 


0.495 


0.2834173 


0.535 


0.2981248 


0.575 


0.3122031 


.496 


.2837932 


.536 


.2984842 


.576 


.3125475 


.497 


.2841686 


.537 


.5988432 


.577 


.3128915 


.498 


.2845436 


.538 


.2992018 


.578 


.3132352 


.499 


.2849181 


.539 


.2995600 


.579 


.3135785 


0.500 


0.2852923 


0.540 


0.2999178 


0.580 


0.3139215 


.501 


.2856660 


.541 


.3002752 


.581 


.3142641 


.502 


.2860392 


.542 


.3006323 


.582 


.3146064 


.503 


.2864121 


.543 


.3009890 


.583 


.3149483 


.504 


.2867845 


.544 


.3013452 


.584 


.3152898 


0.505 


0.2871565 


0.545 


0.3017011 


0.585 


0.3156310 


.506 


.2875281 


.546 


.3020566 


.586 


.3159719 


.507 


.2878992 


.547 


.3024117 


.587 


.3163124 


.508 


.2882700 


.548 


.3027664 


.588 ' 


.3166525 


.509 


.2886403 


.549 


.3031208 


.589 


.3169923 


0.510 


0.2890102 


0.550 


0.3034748 


0.590 


0.3173318 


.511 


.2893797 


.551 


.3038284 


.591 


.3176709 


.512 


.2897487 


.552 


.3041816 


.592 


.3180096 


.513 


.2901174 


.553 


.3045344 


.593 


.3183481 


.514 


.2904856 


.554 


.3048869 


.594 


.3186861 


0.515 


0.2908535 


0.555 


0.3052390 


0.595 


0.3190239 


.516 


.2912209 


.556 


.3055907 


.596 


.3193612 


.517 


.2915879 


.557 


.3059420 


.597 


.3196983 


.518 


.2919545 


.558 


.3062930 


.598 


.3200350 


.519 


.2923207 


.559 


.3066436 


.599 


.3203714 


.520 


.2926864 


.560 


.3069938 


.600 


.3207074 



TABLE III. (See Articles 90, 100.) 



17 



xorz 


S 


f 


X or z 


f 


f 


0.000 


0.0000000 


0.0000000 


0.040 


0.0000936 


0.0000894 


.001 


.0000001 


.0000001 


.041 


.0000984 


.0000938 


.002 


.0000002 


.0000002 


.042 


.0001033 


.0000984 


.003 


.0000005 


.0000005 


.043 


.0001084 


.0001031 


.004 


.0000009 


.0000009 


.044 


.0801135 


.0001079 


0.005 


0.0000014 


0.0000014 


0.045 


0.0001188 


0.0001128 


.006 


.0000021 


.0000020 


.046 


.0001242 


.0001178 


.007 


.0000028 


.0000028 


.047 


.0001298 


.0001229 


.008 


.0000037 


.0000036 


.048 


.0001354 


.0001281 


.009 


.0000047 


.0000046 


.049 


.0001412 


.0001334 


0.010 


0.0000058 


0.0000057 


0.050 


0.0001471 


0.0001389 


.011 


.0000070 


.0000069 


.051 


.0001532 


.0001444 


.012 


.0000083 


.0000082 


.052 


.0001593 


.0001500 


.013 


.0000097 


.0000096 


.053 


.0001656 


.0001558 


.OU 


.0000113 


.0000111 


.054 


.0001720 


.0001616 


0.015 


0.0000130 


0.0000127 


0.055 


0.0001785 


0.0001675 


.016 


.0000148 


.0000145 


.056 


.0001852 


.0001736 


.017 


.0000167 


.0000164 


.057 


.0001920 


.0001798 


.018 


.0000187 


.0000183 


.058 


.0001989 


.0001860 


.019 


.0000209 


.0000204 


.059 


.0002060 


.0001924 


0.020 


0.0000231 


0.0000226 


0.060 


0.0002131 


0.0001988 


.021 


.0000255 


.0000249 


.061 


.0002204 


.0002054 


.022 


.0000280 


.0000273 


.062 


.0002278 


.0002121 


.023 


.0000306 


.0000298 


.063 


.0002354 


.0002189 


.024 


.0000334 


.0000325 


.064 


.0002431 


.0002257 


0.025 


0.0000362 


0.0000352 


0.065 


0.0002509 


0.0002327 


.026 


.0000392 


.0000381 


.066 


.0002588 


.0002398 


.027 


.0000423 


.0000410 


.067 


.0002669 


.0002470 


.028 


.0000455 


.0000441 


.068 


.0002751 


.0002543 


.029 


.0000489 


.0000473 


.069 


.0002834 


.0002617 


0.030 


0.0000523 


0.0000506 


0.070 


0.0002918 


0.0002691 


.031 


.0000559 


.0000539 


.071 


.0003004 


.0002767 


.032 


.0000596 


.0000575 


.072 


.0003091 


.0002844 


.033 


.0000634 


.0000611 


.073 


.0003180 


.0002922 


.034 


.0000674 


.0000648 


.074 


.0003269 


.0003001 


0.035 


0.0000714 


0.0000686 


0.075 


0.0003360 


0.0003081 


.036 


.0000756 


.0000726 


.076 


.0003453 


.0003162 


.037 


.0000799 


.0000766 


.077 


.0003546 


.0003244 


.038 


.0000844 


.0000807 


.078 


.0003641 


.0003327 


.039 


.0000889 


.0000850 


.079 


.0003738 


.0003411 


.040 


.0000936 


.0000894 


.080 


.0003835 


.0003496 



18 



TABLE III. 



X or z 


f 


^ 


X or z 


f 


f 


0.080 


0.0003835 


0.0003496 


0.120 


0.0008845 


0.0007698 


.081 


.0003934 


.0003582 


.121 


.0008999 


.0007822 


.082 


.0004034 


.0003669 


.122 


.0009154 


.0007948 


.083 


.0004136 


.0003757 


.123 


.0009311 


.0008074 


.084 


,0004239 


.0003846 


.124 


.0009469 


.0008202 


0.085 


0.0004343 


0.0003936 


0.125 


0.0009628 


0.0008330 


.086 


.0004448 


.0004027 


.126 


.0009789 


.0008459 


.087 


.0004555 


.0004119 


.127 


.0009951 


.0008590 


.088 


.0004663 


.0004212 


.128 


.0010115 


.0008721 


.089 


.0004773 


.0004306 


.129 


.0010280 


.0008853 


0.090 


0.0004884 


0.0004401 


0.130 


0.0010447 


0.0008986 


.091 


.0004996 


.0004496 


.131 


.0010615 


.0009120 


.092 


.0005109 


.0004593 


.132 


.0010784 


.0009255 


.093 


.0005224 


.0004691 


.133 


.0010955 


.0009390 


.094 


.0005341 


.0004790 


.134 


.0011128 


.0009527 


0.095 


0.0005458 


0.0004890 


0.135 


0.0011301 


0.0009665 


.096 


.0005577 


.0004991 


.136 


.0011477 


.0009803 


.097 


.0005697 


.0005092 


.137 


.0011654 


.0009943 


.098 


.0005819 


.0005195 


.138 


.0011832 


.0010083 


.099 


.0005942 


.0005299 


.139 


.0012012 


.0010224 


0.100 


0.0006066 


0.0005403 


0.140 


0.0012193 


0.0010366 


.101 


.0006192 


.0005509 


.141 


.0012376 


.0010509 


.102 


.0006319 


.0005616 


.142 


.0012560 


.0010653 


.103 


.0006448 


.0005723 


.143 


.0012745 


.0010798 


.104 


.0006578 


.0005832 


.144 


.0012933 


.0010944 


0.105 


0.0006709 


0.0005941 


0.145 


0.0013121 


0.0011091 


.106 


.0006842 


.0006052 


.146 


.0013311 


.0011238 


.107 


.0006976 


.0006163 


.147 


.0013503 


.0011387 


.108 


.0007111 


.0006275 


.148 


.0013696 


.0011536 


.109 


.0007248 


.0006389 


.149 


.0013891 


.0011686 


0.110 


0.0007386 


0.0006503 


0.150 


0.0014087 


0.0011838 


.111 


.0007526 


.0006618 


.151 


.0014285 


.0011990 


.112 


.0007667 


.0006734 


.152 


.0014484 


.0012143 


.113 


.0007809 


.0006851 


.153 


.0014684 


.0012296 


.114 


.0007953 


.0006969 


.154 


.0014886 


.0012451 


0.115 


0.0008098 


0.0007088 


0.155 


0.0015090 


0.0012607 


.116 


.0008245 


.0007208 


.156 


.0015295 


.0012763 


.117 


.0008393 


.0007329 


.157 


.00] 5.502 


.0012921 


.118 


.0008542 


.0007451 


.158 


.0015710 


.0013079 


.119 


.0008693 


.0007574 


.159 


.0015920 


.0013238 


.120 


.0008845 


.0007698 


.160 


.0016131 


.0013398 



TABLE III. 



19 



X or z 


f 


f 


X or z 


^ 


f 


0.160 


0.0016131 


0.0013398 


0.200 


0.0025877 


0.0020507 


.161 


.0016344 


.0013559 


.201 


.0026154 


.0020702 


.162 


.0016559 


.0013721 


.202 


.0026433 


.0020897 


.163 


.0016775 


.0013883 


.203 


.0026713 


.0021094 


.164 


.0016992 


.0014047 


.204 


.0026995 


.0021292 


0.165 


0.0017211 


0.0014211 


0.205 


0.0027278 


0.0021490 


.166 


.0017432 


.0014377 


.206 


.0027564 


.0021689 


.167 


.0017654 


.0014543 


.207 


.0027851 


.0021889 


.168 


.0017878 


.0014710 


.208 


.0028139 


.0022090 


.169 


.0018103 


.0014878 


.209 


.0028429 


.0022291 


0.170 


0.0018330 


0.0015047 


0.210 


0.0028722 


0.0022494 


.171 


.0018558 


.0015216 


.211 


.0029015 


.0022697 


.172 


.0018788 


.0015387 


.212 


.0029311 


.0022901 


.173 


.0019020 


.0015558 


.213 


.0029608 


.0023106 


.174 


.0019253 


.0015730 


.214 


.0029907 


.0023311 


0.175 


0.0019487 


0.0015903 


0.215 


0.0030207 


0.0023518 


.176 


.0019724 


.0016077 


.216 


.0030509 


.0023725 


.177 


.0019961 


.0016252 


.217 


.0030814 


.0023932 


.178 


.0020201 


.0016428 


.218 


.0031119 


.0024142 


.179 


.0020442 


.0016604 


.219 


.0031427 


.0024352 


0.180 


0.0020685 


0.0016782 


0.220 


0.0031736 


0.0024562 


.181 


.0020929 


.0016960 


.221 


.0032047 


.0024774 


.182 


.0021175 


.0017139 


.222 


.0032359 


.0024986 


.183 


.0021422 


.0017319 


.223 


.0032674 


.0025199 


.184 


.0021671 


.0017500 


.224 


.0032990 


.0025412 


0.185 


0.0021922 


0.0017681 


0.225 


0.0033308 


0.0025627 


.186 


.0022174 


.0017864 


.226 


.0033627 


.0025842 


.187 


.0022428 


.0018047 


.227 


.0033949 


.0026058 


.188 


.0022683 


.0018231 


.228 


.0034272 


.0026275 


.189 


.0022941 


.0018416 


.229 


.0034597 


.0026493 


0.190 


0.0023199 


0.0018602 


0.230 


0.0034924 


0.0026711 


.191 


.0023460 


.0018789 


.231 


.0035252 


.0026931 


.192 


.0023722 


.0018976 


.232 


.0035582 


.0027151 


.193 


.0023985 


.0019165 


.233 


.0035914 


.0027371 


.194 


.0024251 


.0019354 


.234 


.0036248 


.0027593 


0.195 


0.0024518 


0.0019544 


0.235 


0.0036584 


0.0027816 


.196 


.0024786 


.0019735 


.236 


.0036921 


.0028039 


.197 


.0025056 


.0019926 


.237 


.0037260 


.0028263 


.198 


.0025328 


.0020119 


.238 


.0037601 


.0028487 


.199 


.0025602 


.0020312 


.239 


.0037944 


.0028713 


.200 


.0025877 


.0020507 


.240 


.0038289 


.0028939 



20 



TABLE III. 



X or z 


f 


f 


X or z 


•f 


f 




0.240 


0.0038289 


0.0028939 


0.270 


0.0049485 


0.0036087 




.241 


.0038635 


.0029166 


.271 


.0049888 


.0036337 




.242 


.0038983 


.0029394 


.272 


.0050292 


.0036587 




.243 


.0039333 


.0029623 


.273 


.0050699 


.0036839 




.244 


.0039685 


.0029852 


.274 


.0051107 


.0037091 




0.245 


0.0040039 


0.0030083 


0.275 


0.0051517 


0.0037344 




.246 


.0040394 


.0030314 


.276 


.0051930 


.0037598 




.247 


.0040752 


.0030545 


.277 


.0052344 


.0037852 




.248 


.0041111 


.0030778 


.278 


.0052760 


.0038107 




.249 


.0041472 


.0031011 


.279 


.0053118 


.0038363 




0.250 


0.0041835 


0.0031245 


0.280 


0.0053598 


0.0038620 




.251 


.0042199 


.0031480 


.281 


.0054020 


.0038877 




.252 


.0042566 


.0031716 


.282 


.0054444 


.0039135 




.253 


.0042934 


.0031952 


.283 


.0054870 


.0039394 




.254 


.0043305 


.0032189 


.284 


.0055298 


.0039654 




0.255 


0.0043677 


0.0032427 


0.285 


0.0055728 


0.0039914 




.256 


.0044051 


.0032666 


.286 


.0056160 


.0040175 




.257 


.0044427 


.0032905 


.287 


.0056594 


.0040437 




.258 


.0044804 


.0033146 


.288 


.0057030 


.0040700 




.259 


.0045184 


.0033387 


.289 


.0057468 


.0040963 




0.260 


0.0045566 


0.0033628 


0.290 


0.0057908 


0.0041227 




.261 


.0045949 


.0033871 


.291 


.0058350 


.0041491 




.262 


.0046334 


.0034114 


.292 


.0058795 


.0041757 




.263 


.0046721 


.0034358 


.293 


.0059241 


■ .0042023 




.264 


.0047111 


.0034603 


.294 


.0059689 


.0042290 




0.265 


0.0047502 


0.0034848 


0.295 


0.0060139 


0.0042557 




.266 


.0047894 


.0035094 


.296 


.0060591 


.0042826 




.267 


.0048289 


.0035341 


.297 


.0061045 


.0043095 




.268 


.0048686 


.0035589 


.298 


.0061502 


.0043364 




.269 


.0049085 


.0035838 


.299 


.0061960 


.0043635 




.270 


.0049485 


.0036087 


.300 


.0062421 


.0043906 





TABLE la. 



21 





ELLIPSE. 


HYPERBOLA. 1 


A 


LogE„ 


Log diff. 


LogE, 


Log diff. 


LogE^ 


Log diff. 


Log E,. 


Log diff 


0.000 


0.0000000 


9.2401 


0.0000000 


9.6378 


0.0000000 


9.2398 


0.0000000 


9.6378 


.001 


.0001738 


.2403 


9.9995656 


.6381 


9.9998263 


.2395 


.0004341 


.6375 


.002 


.0003477 


.2406 


.9991309 


.6384 


.9996528 


.2392 


.0008680 


.6372 


.003 


.0005217 


.2408 


.9986959 


.6386 


.9994794 


.2389 


.0013017 


.6370 


.004 


.0006958 


.2413 


.9982607 


.6389 


.9993061 


.2386 


.0017350 


.6367 


0.005 


0.0008701 


9.2416 


9.9978252 


9.6391 


9.9991329 


9.2383 


0.0021682 


. 9.6365 


.006 


.0010445 


.2418 


.9973895 


.6394 


.9989598 


.2381 


.0026010 


.6362 


.007 


.0012190 


.2420 


.9969535 


.6396 


.9987869 


.2378 


.0030337 


.6360 


.008 


.0013936 


.2423 


.9965173 


.6399 


.9986141 


.2375 


.0034660 


.6357 


.009 


.0015683 


.2428 


.9960807 


.6402 


.9984414 


.2372 


.0038981 


.6354 


0.010 


0.0017432 


9.2430 


9.9956439 


9.6405 


9.9982688 


9.2369 


0.0043299 


9.6352 


.011 


.0019182 


.2433 


.9952068 


.6407 


.9980963 


.2366 


.0047615 


.6349 


.012 


.0020933 


.2435 


.9947695 


.6410 


.9979240 


.2363 


.0051928 


.6347 


.013 


.0022685 


.2438 


.9943319 


.6412 


.9977517 


.2360 


.0056239 


.6344 


.014 


.0024438 


.2443 


.9938941 


.6414 


.9975796 


.2357 


.0060547 


.6342 


0.015 


0.0026193 


9.2445 


9.9934560 


9.6417 


9.9974076 


9.2354 


0.0064853 


9.6339 


.016 


.0027949 


.2448 


.9930176 


.6420 


.9972357 


.2351 


.0069156 


.6336 


.017 


.0029706 


.2453 


.9925789 


.6423 


.9970639 


.2348 


.0073456 


.6334 


.018 


.0031465 


.2455 


.9921400 


.6425 


.9968923 


.2345 


.0077754 


.6331 


.019 


.0033225 


.2458 


.9917008 


.6428 


.9967207 


.2342 


.0082049 


.6329 


0.020 


0.0034986 


9.2460 


9.9912614 


9.6430 


9.99'65493 


9.2339 


0.0086342 


9.6326 


.021 


.0036748 


.2460 


.9908217 


.6433 


.9963780 


.2336 


.0090632 


.6323 


.022 


.0038510 


.2465 


.9903817 


.6436 


.9962068 


.2333 


.0094920 


.6321 


.023 


.0040274 


.2470 


.9899415 


.6438 


.9960357 


.2330 


.0099205 


.6318 


.024 


.0042040 


.2472 


.9895010 


.6441 


.9958648 


.2328 


.0103487 


.6316 


0.025 


0.0043807 


9.2475 


9.9890602 


9.6444 


9.9956939 


9.2325 


0.0107767 


9.6313 


.026 


.0045575 


.2477 


.9886192 


.6446 


.9955232 


.2322 


.0112045 


.6311 


.027 


.0047344 


.2480 


.9881779 


.6449 


.9953526 


.2319 


.0116320 


.6308 


.028 


.0049114 


.2485 


.9877363 


.6452 


.9951821 


.2316 


.0120592 


.6306 


.029 


.0050886 


.2487 


.9872945 


.6454 


.9950117 


.2313 


.0124862 


.6303 1 

1 


0.030 


0.0052659 


9.2490 


9.9868524 


9.6457 


9.9948414 


9.2310 


0.0129130 


9.6301 ! 


.031 


.0054433 


.2494 


.9864100 


.6459 


.9946712 


.2307 


.0133395 


.6298 i 


.032 


.0056209 


.2497 


.9859674 


.6462 


.9945012 


.2304 


.0137657 


.6295 


.033 


.0057986 


.2499 


.9855245 


.6465 


.9943313 


.2301 


.0141917 


.6293 


.034 


.0059764 


.2502 


.9850813 


.6468 


.9941615 


.2298 


.0146175 


.6290 


0.035 


0.0061543 


9.2504 


9.9846378 


9.6471 


9.9939918 


9.2295 


0.0150430 


9.6288 


.036 


.0063323 


.2509 


.9841940 


.6474 


.9938222 


.2292 


.0154683 


.6285 1 


.037 


.0065105 


.2512 


.9837499 


.6476 


.9936528 


.2290 


.0158933 


.6283 1 


.038 


.0066888 


.2514 


.9833056 


.6478 


.9934834 


.2287 


.0163180 


.6280 1 


.039 


.0068672 


.2516 


.9828610 


.6481 


.9933142 


.2284 


.0167426 


.6278 


.040 


.0070457 


.2519 


.9824161 


.6484 


.9931450 


.2281 


.0171668 


.6275 



22 



ABLE la. 





ELLIPSE. 


HYPERBOLA. 


A 


Log E„ 


Log diflf. 


LogE, 


Log diff. 


LogE^ 


Log diflf. 


Log E,. 


Log DiflT. 


0.040 


0.0070457 


9.2519 


9.9824161 


9.6484 


9.9931450 


9.2281 


0.0171668 


9.6275 


.041 


.0072243 


.2524 


.9819709 


.6487 


.9929760 


.2278 


.0175908 


.6273 


.042 


.0074031 


.2526 


.9815255 


.6489 


.9928071 


.2275 


.0180146 


.6270 


.043 


.0075820 


.2531 


.9810798 


.6492 


.9',l2(;;>83 


.2272 


.0184381 


.6267 


.044 


.0077611 


.2533 


.9806339 


.6494 


.9924696 


.2269 


.0188614 


.6265 


0.045 


0.0079403 


9.2536 


9.9801877 


9.6497 


9.9923010 


9.2266 


0.0192844 


9.6262 


.046 


.0081196 


.2538 


.9797412 


.6500 


.9921325 


.2263 


.0197072 


.6260 


.047 


.0082990 


.2543 


.9792944 


.6502 


.9919642 


.2260 


.0201297 


.6257 


.048 


.0084786 


.2546 


.9788474 


.6505 


.9917960 


.2258 


.0205520 


.6255 


.049 


.0086583 


.2548 


.9784001 


.6508 


.9916279 


.2255 


.0209740 


.6252 


0.050 


0.0088381 


9.2550 


9.9779525 


9.6511 


9.9914599 


9.2252 


0.0213958 


9.6250 


.051 


.0090180 


.2555 


.9775046 


.6514 


.9912920 


.2249 


.0218174 


.6247 


.052 


.0091981 


.2558 


.9770564 


.6516 


.9911242 


.2246 


.0222387 


.6245 


.053 


.0093783 


.2560 


.9766079 


.6519 


.9909565 


.2243 


.0226597 


.6242 


.054 


.0095586 


.2565 


.9761592 


.6521 


.9907890 


.2240 


.0230805 


.6240 


0.055 


0.0097391 


9.2567 


9.9757102 


9.6524 


9.990G215 


9.2237 


0.0235011 


9.6237 


.056 


.0099197 


.2570 


.9752609 


.6527 


.9904542 


.2235 


.0239214 


.6235 ! 


.057 


.0101004 


.2572 


.9748113 


.6529 


.9902869 


.2232 


.0243415 


.6232 1 


.058 


.0102812 


.2577 


.9743615 


.6532 


.91101198 


.2229 


.0247614 


.6230 


.059 


.0104622 


.2579 


.9739114 


.6535 


.9)S',)9528 


.2226 


.0251810 


.6227 ; 


0.060 


0.0106433 


9.2582 


9.9734611 


9.6538 


9.9897859 


9.2223 


0.0256003 


9.6225 


.061 


.0108245 


.2584 


.9730103 


.6541 


.9896191 


.2220 


.0260194 


.6222 


.062 


.0110058 


.2589 


.9725593 


.6543 


.9894525 


.2217 


.0264383 


.6220 ; 


.063 


.0111873 


.2591 


.9721080 


.6546 


.9.S'.)2S5:) 


.2214 


.0268570 


.6217 


.064 


.0113689 


.2594 


.9716565 


.6548 


.9891195 


.2211 


.0272753 


.6215 


0.065 


0.0115506 


9.2598 


9.9712047 


9.6551 


9.9889531 


9.2208 


0.0276935 


9.6212 


.066 


.0117325 


.2601 


.9707526 


.6554 


.9887 869 


.2206 


.0281114 


.6210 


.067 


.0119145 


.2603 


.9703002 


.6557 


.9SSG208 


.2203 


.0285291 


.6207 


.068 


.0120966 


.2606 


.9698475 


.6560 


.9884548 


.2200 


.0289465 


.6205 


.069 


.0122788 


.2610 


.9693945 


.6562 


.9882889 


.2197 


.0293637 


.6202 


0.070 


0.0124612 


9.2613 


9.9689413 


9.6565 


9.9881231 


9.2194 


0.0297807 


9.6200 


.071 


.0126437 


.2617 


.9684878 


.6567 


.9879574 


.2191 


.0301974 


.6197 


.072 


.0128264 


.2620 


.9680340 


.6570 


.9877918 


.2189 


.0306139 


.6195 


.073 


.0130092 


.2622 


.9675799 


.6573 


.9876263 


.2186 


.0310301. 


.6192 


.074 


.0131921 


.2625 


.9671255 


.6576 


.9874610 


.2183 


.0314461 


.6190 


0.075 


0.0133751 


9.2629 


9.9666708 


9.6578 


9.9872957 


9.2180 


0.0318618 


9.6187 


.076 


.0135583 


.2632 


.9662159 


.6581 


.9871306 


.2177 


.0322773 


.6185 


.077 


.0137416 


.2634 


.9657606 


.6584 


.9869655 


.2174 


.0326926 


.6182 


.078 


.0139250 


.2638 


.9653051 


.6587 


.9868006 


.2172 


.0331076 


.6180 ' 


.079 


.0141086 


.2641 


.9648492 


.6590 


.9866358 


.2169 


.0335224 


.6177 


.080 


.0142923 


.2643 


.9643931 


.6592 


.9864711 


.2166 


.0339370 


.6175 

i 



TABLE la. 



23 





ELLIPSE. 


HYPERBOLA. 


A 


LogE„ 


Log diff. 


LogE^ 


Log diff. 


LogE, 


Log diff. 


Log E,. 


Log Diff. 


0.080 


0.0142923 


9.2643 


9.9643931 


9.6592 


9.9864711 


9.2166 


0.0339370 


9.6175 ' 


.081 


.0144761 


.2646 


.9639367 


.6595 


.9863065 


.2163 


.0343513 


.6172 


.082 


.0146601 


.2649 


.9634800 


.6598 


.9861420 


.2160 


.0347654 


.6170 


.083 


.0148442 


.2652 


.9630230 


.6600 


.9859776 


.2157 


.0351793 


.6167 


.084 


.0150284 


.2655 


.9625657 


.6603 


.9858133 


.2155 


.0355930 


.6165 


0.085 


0.0152128 


9.2659 


9.9621081 


9.6606 


9.9856491 


9.2152 


0.0360064 


9.6163 


.086 


.0153973 


.2662 


.9616503 


.6609 


.9854850 


.2149 


.0364196 


.6160 


.087 


.0155819 


.2665 


.9611922 


.6611 


.9853210 


.2146 


.0368325 


.6158 


.088 


.0157667 


.2668 


.9607337 


.6614 


.9851572 


.2143 


.0372452 


.6155 


.089 


.0159516 


.2671 


.9602749 


.6617 


.9849934 


.2140 


.0376577 


.6153 


0.090 


0.0161367 


9.2674 


9.9598159 


9.6620 


9.9848298 


9.2138 


0.0380699 


9.6150 


.091 


.0163218 


.2677 


.9593566 


.6623 


.9846663 


.2135 


.0384819 


.6148 


.092 


.0165071 


.2680 


.9588970 


.6625 


.9845028 


.2132 


.0388937 


.6145 


.093 


.0166925 


.2684 


.9584371 


.6628 


.9843395 


.2129 


.0393052 


.6143 


.094 


.0168781 


.2687 


.9579769 


.6631 


.9841763 


.2126 


.0397165 


.6141 


0.095 


0.0170638 


9.2690 


9.9575164 


9.6634 


9.9840132 


9.2123 


0.0401276 


9.6138 


■ .096 


.0172497 


.2693 


.9570556 


.6636 


.9838502 


.2121 


.0405385 


.6136 


.097 


.0174357 


.2696 


.9565945 


.6639 


.9836873 


.2118 


.0409491 


.6133 


.098 


.0176218 


.2700 


.9561331 


.6642 


.9835245 


.2115 


.0413595 


.6131 


.099 


.0178081 


.2703 


.9556714 


.6645 


.9833618 


.2112 


.0417696 


.6128 


0.100 


0.0179945 


9.2706 


9.9552095 


9.6648 


9.9831992 


9.2109 


0.0421796 


9.6126 


.101 


.0181810 


.2708 


.9547472 


.6650 


.9830367 


.2107 


.0425893 


.6123 


.102 


.0183677 


.2712 


.9542847 


.6653 


.9828743 


.2104 


.0429988 


.6121 


.103 


.0185545 


.2715 


.9538218 


.6656 


.9827121 


.2101 


.0434080 


.6118 


.104 


.0187414 


.2718 


.9533586 


.6659 


.9825499 


.2098 


.0438170 


.6116 


0.105 


0.0189285 


9.2722 


9.9528951 


9.6662 


9.9823879 


9.2095 


0.0442258 


9.6114 


.106 


.0191157 


.2725 


.9524314 


.6664 


.9822259 


.2093 


.0446343 


.6111 


.107 


.0193030 


.2728 


.9519673 


.6666 


.9820641 


.2090 


.0450426 


.6109 


.108 


.0194905 


.2731 


.9515030 


.6670 


.9819023 


.2087 


.0454507 


.6'106 


.109 


.0196781 


.2734 


.9510383 


.6673 


.9817407 


.2084 


.0458585 


.6104 


0.110 


0.0198659 


9.2738 


9.9505734 


9.6676 


9.9815791 


9.2081 


0.0462661 


9.6101 


.111 


.0200538 


.2741 


.9501081 


.6678 


.9814177 


.2079 


.0466735 


.6099 


.112 


.0202418 


.2744 


.9496425 


.6681 


.9812563 


.2076 


.0470807 


.6096 


.113 


.0204300 


.2747 


.9491766 


.6684 


.9810951 


.2073 


.0474876 


.6094 


.114 


.0206183 


.2750 


.9487105 


.6687 


.9809340 


.2070 


.0478943 


.6092 


0.115 


0.0208067 


9.2754 


9.9482440 


9.6690 


9.9807730 


9.2067 


0.0483008 


9.6089 


.116 


.0209953 


.2757 


.9477772 


.6692 


.9806121 


.2065 


.0487071 


.6087 


.117 


.0211840 


.2760 


.9473101 


.6695 


.9804513 


.2062 


.0491131 


.6084 


.118 


.0213729 


.2763 


.9468428 


.6698 


.9802905 


.2059 


.0495189 


.6082 


.119 


.0215619 


.2767 


.9463751 


.6701 


.9801299 


.2056 


.0499245 


.6080 


.120 


.0217511 


.2770 


.9459071 


.6704 


.9799694 


.2054 


.0503298 


.6077 



24 



TABLE Ic 





ELLIPSE. 


HYPERBOLA. 


A 


LogE„ 


Log diflf. 


Log E^ 


Log diff. 


Log E^ 


Log diff. 


Log E,. 


Log Diff. 


0.120 


0.0217511 


9.2770 


9.9459071 


9.6704 


9.9799694 


9.2054 


0.0503298 


9.6077 


.121 


.0219404 


.2773 


.9454388 


.6707 


.9798090 


.2051 


.0507349 


.6075 


.122 


.0221298 


.2776 


.9449702 


.6709 


.'.t7'.t(M87 


.2048 


.0511399 


.6072 


.123 


.0223193 


.2779 


.9445013 


.6712 


.97'.) I.S85 


.2045 


.0515446 


.6070 


.124 


.0225091 


.2783 


.9440321 


.6715 


.9793284 


.2043 


.0519490 


.6068 


0.125 


0.0226990 


9.2786 


9.9435626 


9.6718 


9.9791684 


9.2040 


0.0523533 


5.6065 


.126 


.0228889 


.2789 


.9430927 


.6721 


.9790085 


.2037 


.0527573 


.6063 


.127 


.0230791 


.2792 


.9426226 


.6724 


.9788487 


.2034 


.0531611 


.6061 


.128 


.0232693 


.2795 


.9421521 


.6727 


.9786890 


.2032 


.0535647 


.6058 


.129 


.0234597 


.2799 


.9416813 


.6729 


.9785294 


.2029 


.0539681 


.6056 : 


0.130 


0.0236503 


9.2802 


9.9412103 


9.6732 


9.9783699 


9.2026 


0.0543712 


).6053 


.131 


.0238410 


.2805 


.9407389 


.6735 


.9782105 


.2023 


.0547741 


.6051 


.132 


.0240318 


.2808 


.9402672 


.6738 


.9780512 


.2021 


.0551768 


.6049 


.133 


.0242228 


.2812 


.9397952 


.6741 


.9778920 


.2018 


.0555793 


.6046 


.134 


.0244139 


.2815 


.9393229 


.6744 


.9777329 


.2015 


.0559816 


.6044 


0.135 


0.0246052 


9.2818 


9.9388503 


9.6747 


9.9775739 


9.2012 


0.0563836 < 


).6041 


.136 


.0247966 


.2822 


.9383773 


.6749 


.9774150 


.2010 


.0567854 


.6039 


.137 


.0249882 


.2825 


.9379041 


.6752 


.9772562 


.2007 


.0571870 


.6037 


.138 


.0251799 


.2828 


.9374305 


.6755 


.9770975 


.2004 


.0575884 


.6034 


.139 


.0253717 


.2831 


.9369567 


.6758 


.9769390 


.2001 


.0579895 


.6032 


0.140 


0.0255637 


9.2834 


9.9364824 


9.6761 


9.9767805 


9.1998 


0.0583904 f 


).6029 


.141 


.0257558 


.2838 


.9360079 


.6764 


.9766221 


.1996 


.0587911 


.6027 


.142 


.0259481 


.2841 


.9355331 


.6767 


.9764638 


.1993 


.0591916 


.6025 


.143 


.0261405 


.2844 


.9350580 


.6770 


.9763057 


.1990 


.0595919 


.6022 


.144 


.0263331 


.2848 


.9345825 


.6773 


.9761476 


.1988 


.0599919 


.6020 


0.145 


0.0265258 


9.2851 


9.9341067 


9.6775 


9.9759896 


9.1985 


0.0603917 { 


).6018 


.146 


.0267187 


.2854 


.9336307 


.6778 


.9758317 


.1982 


.0607913 


.6015 


.147 


.0269117 


.2857 


.9331543 


.6781 


.9756739 


.1979 


.0611907 


.6013 


.148 


.0271048 


.2861 


.9326775 


.6784 


.9755162 


.1977 


.0615899 


.6010 


.149 


.0272981 


.2864 


.9322005 


.6787 


.9753586 


.1974 


.0619888 


.6008 


0.150 


0.0274915 


9.2867 


9.9317231 


9.6790 


9.9752011 


9.1971 


0.0623876 i 


.6006 


.151 


.0276851 


.2871 


.9312455 


.6793 


.9750437 


.1969 


.0627861 


.6003 


.152 


.0278789 


.2874 


.9307675 


.6796 


.9748864 


.1966 


.0631844 


.6001 


.153 


.0280728 


.2877 


.9302892 


.6798 


.9747292 


.1963 


.0635825 


.5999 


. .154 


.0282668 


.2880 


.9298106 


.6801 


.9745721 


.1960 


.0639804 


.5996 


0.155 


0.0284610 


9.2884 


9.9293317 


9.6804 


9.9744151 


9.1958 


0.0643780 i 


.5994 


.156 


.0286553 


.2887 


.9288524 


.6807 


.9742582 


.1955 


.0647755 


.5992 


.157 


.0288498 


.2890 


.9283728 


.6810 


.9741014 


.1952 


.0651727 


.5989 


.158 


.0290444 


.2893 


.9278929 


.6813 


.9739447 


.1949 


.0655697 


.5987 


.159 


.0292392 


.2897 


.9274127 


.6816 


.9737881 


.1946 


.0659665 


.5985 


.160 


.0294341 


.2900 


.9269321 


.6819 


.9736316 


.1944 


.0663631 


.5982 



TABLE la. 



25 







ELLIPSE. 




HYPERBOLA. 


A 


LosE^ 


Log diflf. 


LogE, 


Log difF. 


LogE^ 


Log diff. 


LogE.. 


Log diff. 


0.160 


0.0294341 


9.2900 


9.9269321 


9.6819 


9.9736316 


9.1944 


0.0663631 


9.5982 


.161 


.0296292 


.2903 


.9264512 


.6822 


.9734752 


.1941 


.0667595 


.5980 


.162 


.0298243 


.2906 


.9259700 


.6825 


.9733189 


.1938 


.0671556 


.5978 


.163 


.0300197 


.2910 


.9254885 


.6828 


.9731627 


.1936 


.0675516 


.5975 i 


.164 


.0302152 


.2913 


.9250067 


.6831 


.9730066 


.1933 


.0679473 


.5973 


0.165 


0.0304109 


9.2916 


9.9245245 


9.6833 


9.9728506 


9.1930 


0.0683428 


9.5971 


.166 


.0306067 


.2920 


.9240421 


.6836 


.9726947 


.1928 


.0687381 


.5968 


.167 


.0308026 


.2923 


.9235592 


.6839 


.9725389 


.1925 


.0691332 


.5966 


.168 


.0309987 


.2926 


.9230761 


.6842 


.9723831 


.1922 


.0695281 


.5963 


.169 


.0311949 


.2930 


.9225926 


.6845 


.9722275 


.1920 


.0699228 


.5961 


0.170 


0.0313913 


9.2933 


9.9221089 


9.6848 


9.9720719 


9.1917 


0.0703172 


9.5959 


.171 


.0315879 


.2936 


.9216247 


.6851 


.9719165 


.1914 


.0707114 


.5956 


.172 


.0317846 


.2940 


.9211403 


.6854 


.9717611 


.1912 


.0711055 


.5954 


.173 


.0319815 


.2943 


.9206555 


.6857 


.9716059 


.1909 


.0714993 


.5952 


.174 


.0321784 


.2946 


.9201704 


.6860 


.9714507 


.1906 


.0718929 


.5949 


0.175 


0.0323756 


9.2950 


9.9196850 


9.6863 


9.9712957 


9.1904 


0.0722863 


9.5947 


.176 


.0325729 


.2953 


.9191992 


.6866 


.9711407 


.1901 


.0726795 


.5945 


.177 


.0327704 


.2956 


.9187131 


.6869 


.9709859 


.1898 


.0730724 


.5942 


.178 


.0329680 


.2960 


.9182266 


.6872 


.9708311 


.1895 


.0734652 


.5940 


.179 


.0331657 


.2963 


.9177399 


.6875 


.9706764 


.1893 


.0738578 


.5938 


0.180 


0.0333636 


9.2966 


9.9172528 


9.6878 


9.9705218 


9.1890 


0.0742501 


9.5935 


.181 


.0335617 


.2970 


.9167654 


.6881 


.9703673 


.1887 


.0746422 


.5933 


.182 


.0337599 


.2973 


.9162776 


.6884 


.9702129 


.1885 


.0750341 


.5931 


.183 


.0339582 


.2977 


.9157895 


.6886 


.9700587 


.1882 


.0754259 


.5928 


.184 


.0341568 


.2980 


.9153011 


.6889 


.9699045 


.1879 


.0758173 


.5926 


0.185 


0.0343555 


9.2983 


9.9148123 


9.6892 


9.9697504 


9.1877 


0.0762086 


9.5924 


.186 


.0345543 


.2987 


.9143232 


.6895 


.9695964 


.1874 


.0765997 


.5922 


.187 


.0347533 


.2990 


.9138338 


.6898 


.9694425 


.1871 


.0769906 


.5919 


.188 


.0349524 


.2993 


.9133441 


.6901 


.9692887 


.1869 


.0773812 


.5917 


.189 


.0351517 


.2997 


.9128540 


.6904 


.9691350 


.1866 


.0777717 


.5915 


0.190 


0.0353511 


9.3000 


9.9123635 


9.6907 


9.9689813 


9.1863 


0.0781619 


9.5912 


.191 


.0355507 


.3003 


.9118727 


.6910 


.9688278 


.1861 


.0785520 


.5910 


.192 


.0357505 


.3007 


.9113816 


.6913 


.9686743 


.1858 


.0789418 


.5908 


.193 


.0359504 


.3010 


.9108901 


.6916 


.9685210 


.1855 


.0793315 


.5906 


.194 


.0361505 


.3014 


.9103983 


.6919 


.9683678 


.1853 


.0797209 


.5903 


0.195 


0.0363507 


9.3017 


9.9099062 


9.6922 


9.9682146 


9.1850 


0.0801102 


9.5901 


.196 


.0365511 


.3-020 


.9094138 


.6925 


.9680615 


.1847 


.0804992 


.5899 


.197 


.0367516 


.3024 


.9089210 


.6928 


.9679086 


.1845 


.0808881 


.5896 


.198 


.0369523 


.3027 


.9084278 


.6931 


.9677557 


.1842 


.0812767 


.5894 


.199 


.0371532 


.3031 


.9079343 


.6934 


.9676029 


.1839 


.0816651 


.5892 


.200 


.0373542 


.3034 


.9074405 


.6937 


.9674502 


.1837 


.0820533 


.5889 



26 



TABLE la. 





ELLIPSE. 


HYPERBOLA. 


A 


Lo- E,,. 
0.0373542 


Log diff. 


Log E, 


Log difl-. 


Log E,. 


Log diff. 


Log E,. 


Log DiflF. 


0.200 


9.3034 


9.9074405 


9.6937 


9.9674502 


9.1837 


0.0820533 


9.5889 


.201 


.0375554 


.3037 


.9069463 


.6940 


.9672976 


.1834 


.0824413 


.5887 


.202 


.0377567 


.3041 


.9064518 


.6943 


.9671451 


.1831 


.0828291 


.5885 


.203 


.0379582 


.3044 


.9059569 


.6946 


.9669927 


.1829 


.0832166 


.5882 


.204 


.0381598 


.3047 


.9054617 


.6949 


.9668404 


.1826 


.0836040 


.5880 


0.205 


0.0383616 


9.3051 


9.9049662 


9.6952 


9.9666882 


9.1823 


0.0839911 


9.5878 


.206 


.0385635 


.3054 


.9044703 


.6955 


.9665361 


.1821 


.0843781 


.5876 


.207 


.0387656 


.3058 


.9039741 


.6958 


.9G(;;->84i 


.1818 


.0847649 


.5873 


.208 


.0389679 


.3061 


.9034775 


.6961 


.:)G62321 


.1815 


.0851514 


.5871 


.209 


.0391703 


.3065 


.9029806 


.6964 


.9660803 


.1813 


.0855377 


.5869 


0.210 


0.0393729 


9.3068 


9.9024833 


9.6967 


9.9659285 


9.1810 


0.0859239 


9.5867 


.211 


.0395757 


.3071 


.9019857 


.6970 


.9657768 


.1808 


.0863099 


.5864 


.212 


.0397786 


.3075 


.9014877 


.6974 


.9656253 


.1805 


.0866956 


.5862 


.213 


.0399817 


.3078 


.9009894 


.6977 


.9654738 


.1802 


.0870812 


.5860 


.214 


.0401849 


.3081 


.9004907 


.6980 


.9653224 


.1800 


.0874665 


.5858 


0.215 


0.0403883 


9.3085 


9.8999917 


9.6983 


9.9651711 


9.1797 


0.0878517 


9.5855 


.216 


.0405918 


.3088 


.8994924 


.6986 


.9650199 


.1795 


.0882367 


.5853 


.217 


.0407955 


.3092 


.8989927 


.6989 


.9648687 


.1792 


.0886214 


.5851 


.218 


.0409994 


.3095 


.8984927 


.6992 


.9647177 


.1789 


.0890060 


.5849 


.219 


.0412034 


.3099 


.8979923 


.6995 


.9645667 


.1787 


.0893903 


.5846 


0.220 


0.0414076 


9.3102 


9.8974915 


9.6998 


9.9644159 


9.1784 


0.0897745 


9.5844 


.221 


.0416120 


.3106 


.8969904 


.7001 


.9642651 


.1782 


.0901585 


.5842 


.222 


.0418165 


.3109 


.8964889 


.7004 


.9641145 


.1779 


.0905422 


.5839 


.223 


.0420211 


.3112 


.8959881 


.7007 


.9639639 


.1776 


.0909258 


.5837 


.224 


.0422260 


.3116 


.8954849 


.7010 


.9638134 


.1774 


.0913091 


.5835 


0.225 


0.0424310 


9.3119 


9.8949824 


9.7013 


9.9636630 


9.1771 


0.0916923 


9.5833 


.226 


.0426362 


.3123 


.8944795 


.7016 


.9635127 


.1768 


.0920753 


.5830 


.227 


.0428415 


.3127 


.8939762 


.7019 


.9633625 


.1766 


.0924580 


.5828 


.228 


.0430470 


.3130 


.8934726 


.7022 


.9632123 


.1763 


.0928405 


.5826 


.229 


.0432527 


.3133 


.8929687 


.7025 


.9630623 


.1760 


.0932229 


.5823 


0.230 


0.0434585 


9.3137 


9.8924644 


9.7028 


9.9629124 


9.1758 


0.0936050 


9.5821 


.231 


.0436645 


.3140 


.8919597 


.7031 


.9627625 


.1755 


.0939870 


.5819 


.232 


.0438707 


.3144 


.8914547 


.7035 


.9626128 


.1752 


.0943687 


.5817 


.233 


.0440770 


.3147 


.8909493 


.7038 


.9624631 


.1750 


.0947503 


.5814 


.234 


.0442835 


.3151 


.8904436 


.7041 


.9623136 


.1747 


.0951317 


.5812 


0.235 


0.0444902 


9.3154 


9.8899375 


9.7044 


9.9621641 


9.1745 


0.0955128 


9.5810 


.236 


.0440970 


.3158 


.8894310 


.7047 


.9620147 


.1742 


.0958938 


.5808 


.237 


.0419040 


.3161 


.8889242 


.7050 


.9618654 


.1740 


.0962745 


.5806 


.238 


.0451111 


.3165 


.8884170 


.7053 


.9617162 


.1737 


.0966551 


.5803 


.239 


.0453184 


.3168 


.8879094 


.7056 


.9615670 


.1734 


.0970355 


.5801 


.240 


.0455259 


.3171 


.8874015 


.7059 


.9614180 


.1732 


.0974157 


.5799 



TABLE Ic 



27 





ELLIPSE. 


HYPERBOLA. 


A 


l^og E„ 


Log diif. 


LogE, 


Log difF. 


LogE„ 


Log diff. 


LogE,. 


Log Diff. 


0.240 


0.0455259 


9.3171 


9.8874015 


9.7059 


9.9614180 


9.1732 


0.0974157 


9.5799 


.241 


.0457335 


.3175 


.8868932 


.7063 


.9612690 


.1729 


.0977957 


.5797 


.242 


.0459413 


.3179 


.8863846 


.7066 


.9611202 


.1727 


.0981755 


.5794 


.243 


.0461493 


.3182 


.8858756 


.7069 


.9609714 


.1724 


.0985551 


.5792 


.244 


.0463575 


.3186 


.8853663 


.7072 


.9608227 


.1722 


.0989345 


.5790 


0.245 


0.0465658 


9.3189 


9.8848566 


9.7075 


9.9606741 


9.1719 


0.0993137 


9.5788 


.246 


.0467743 


.3193 


.8843465 


.7078 


.9605256 


.1716 


.0996927 


.5786 


.247 


.0469830 


.3196 


.8838360 


.7081 


.9603771 


.1714 


.1000716 


.5783 


.248 


.0471918 


.3200 


.8833252 


.7084 


.9602288 


.1711 


.1004502 


.5781 


.249 


.0474008 


.3203 


.8828140 


.7087 


.9600805 


.1709 


.1008287 


.5779 


0.250 


0.0476099 


9.3207 


9.8823025 


9.7090 


9.9599324 


9.1706 


0.1012069 


9.5777 


.251 


.0478193 


.3210 


.8817906 


.7094 


.9597843 


.1704 


.1015850 


.5775 


.252 


.0480288 


.3214 


.8812783 


.7097 


.9596363 


.1701 


.1019628 


.5772 


.253 


.0482385 


• .3217 


.8807657 


.7100 


.9594884 


.1698 


.1023405 


.5770 


.254 


.0484483 


.3221 


.8802526 


.7103 


.9593406 


.1696 


.1027180 


.5768 


0.255 


0.0486583 


9.3224 


9.8797392 


9.7106 


9.9591929 


9.1693 


0.1030953 


9.5766 


.256 


.0488685 


.3226 


.8792254 


.7109 


.9590453 


.1691 


.1034724 


.5763 


.257 


.0490788 


.3231 


.8787113 


.7112 


.9588977 


.1688 


.1038493 


.5761 


.258 


.0492893 


.3235 


.8781968 


.7116 


.9587502 


.1685 


.1042259 


.5759 


.259 


.0495000 


.3238 


.8776819 


.7119 


.9586029 


.1683 


.1046024 


.5756 


0.260 


0.0497109 


9.3242 


9.8771666 


9.7122 


9.9584556 


9.1680 


0.1049787 


9.5754 


.261 


.0499219 


.3245 


.8766510 


.7125 


.9583084 


.1678 


.1053548 


.5752 


.262 


.0501331 


.3249 


.8761350 


.7128 


.9581613 


.1675 


.1057308 


.5750 


.263 


.0503445 


.3252 


.8756186 


.7131 


.9580143 


.1673 


.1061065 


.5748 


.264 


.0505560 


.3256 


.8751019 


.7134 


.9578673 


.1670 


.1064821 


.5746 


0.265 


0.0507677 


9.3260 


9.8745848 


9.7137 


9.9577205 


9.1668 


0.1068574 


9.5743 


.266 


.0509796 


.3263 


.8740673 


.7141 


.9575737 


.1665 


.1072326 


.5741 


.267 


.0511917 


.3267 


.8735495 


.7144 


.9574270 


.1662 


.1076076 


.5739 


.268 


.0514040 


.3270 


.8730312 


.7147 


.9572804 


.1660 


.1079824 


.5737 


.269 


.0516164 


.3274 


'.8725126 


.7150 


.9571339 


.1657 


.1083570 


.5735 


0.270 


0.0518290 


9.3277 


9.8719936 


9.7153 


9.9569875 


9.1655 


0.1087314 


9.5733 


.271 


.0520418 


.3281 


.8714742 


.7157 


.9568412 


.1652 


.1091056 


.5730 


.272 


.0522547 


.3284 


.8709544 


.7160 


.9566949 


.1650 


.1094797 


.5728 


.273 


.0524678 


.3288 


.8704343 


.7163 


.9565487 


.1647 


.1098536 


.5726 


.274 


.0526811 


.3292 


.8699137 


.7166 


.9564027 


.1644 


.1102272 


.5724 


0.275 


0.0528946 


9.3295 


9.8693928 


9.7169 


9.9562567 


9.1642 


0.1106007 


9.5722 


.276 


.0531082 


.3299 


.8688715 


.7173 


.9561108 


.1639 


.1109740 


.5719 


.277 


.0533220 


.3303 


.8683498 


.7176 


.9559650 


.1637 


.1113471 


.5717 


.278 


.0535360 


.3306 


.8678278 


.7179 


.9558193 


.1634 


.1117200 


.5715 


.279 


.0537502 


.3310 


.8673053 


.7182 


.9556736 


.1632 


.1120927 


.5713 


.280 


.0539646 


.3313 


.8667825 


.7185 


.9555281 


.1629 


.1124652 


.5710 



28 



TABLE la. 





ELLIPSE. 




HYPERBOLA. 




A 


LogE„ 


Log diff. 


Log E, 


Log diff. 


Log E^ 


Log diff. 


LogE,.. 


Log Diff. 


0.280 


0.0539646 


9.3313 


9.8667825 


9.7185 


9.9555281 


9.1629 


0.1124652 


9.5710 


.281 


.0541791 


.3317 


.8662593 


.7188 


.9553826 


.1627 


.1128375 


.5708 


.282 


.0543939 


.3320 


.8657357 


.7192 


.9552372 


.1624 


.1132097 


.5707 


.283 


.0546087 


.3324 


.8652117 


.7195 


.9550919 


.1622 


.1135817 


.5704 


.284 


.0548238 


.3327 


.8646873 


.7198 


.9549467 


.1619 


.1139534 


.5701 


0.285 


0.0550390 


9.3331 


9.8641625 


9.7201 


9.9548015 


9.1617 


0.1143250 


9.5699 


.286 


.0552546 


• .3335 


.8636374 


.7204 


.9546564 


.1614 


.1146964 


.5698 


.287 


.0554700 


.3338 


.8631118 


.7208 


.9545115 


.1612 


.1150677 


.5695 


.288 


.0556858 


.3342 


.8625859 


.7211 


.9543666 


.1609 


.1154387 


.5693 


.289 


.0559018 


.3345 


.8620596 


.7214 


.'9542218 


.1606 


.1158096 


.5691 


0.290 


0.0561179 


9.3349 


9.8615329 


9.7217 


9.9540771 


9.1604 


0.1161803 


9.5689 


.291 


.0563342 


.3353 


.8610058 


.7221 


.9539325 


.1601 


.1165508 


.5687 


.292 


.0565507 


.3356 


.8604783 


.7224 


.9537879 


.1599 


.1169211 


.5685 


.293 


.0567674 


.3360 


.8599504 


.7227 


.9536435 


.1596 


.1172913 


.5683 


.294 


.0569842 


.3364 


.8594221 


.7230 


.9534991 


.1594 


.1176612 


.5680 


0.295 


0.0572013 


9.3367 


9.8588935 


9.7233 


9.9533548 


9.1591 


0.1180310 


9.5678 


.296 


.0574185 


.3371 


.8583644 


.7236 


.9532106 


.1589 


.1184006 


.5675 1 


.297 


.0576359 


.3375 


.8578349 


.7240 


.9530665 


.1586 


.1187699 


.5673 


.298 


.0578535 


.3379 


.8573051 


.7243 


.9529224 


.1584 


.1191391 


.5671 


.299 


.0580713 


.3383 


.8567748 


.7246 


.9527785 


.1581 


.1195081 


.5668 


.300 


.0582893 


.3387 


.8562442 


.7249 


.9526346 


.1578 


0.1198768 


9.5666 



TABLE Ua. 



29 



ro- 


Vo- 


LogAi. 


LogAj. 


Log A3. 





6 0.00 


4-3.7005216 


—0.00000 


—9.695 


2 


2 47 11.83 


3.7000079 


0.47160 


9.691 


4 


5 34 0.00 


3.6984710 


0.76930 


9.681 


6 


8 20 1.19 


3.6959236 


0.93987 


9.664 


8 


11 4 52.82 


3.6923863 


1.05702 


9.641 


10 


J 3 48 13.31 


+3.6878872 


—1.14430 


—9.610 


12 


16 29 42.39 


3.6824613 


1.21171 


9.571 


14 


19 9 1.36 


3.6761493 


1.26497 


9.525 


16 


21 45 53.23 


3.6689972 


1.30744 


9.470 


18 


24 20 2.89 


3.6610547 


1.34135 


9.405 


20 


26 51 17.15 


+3.6523748 


—1.36825 


—9.329 


22 


29 19 24.78 


3.6430121 


1.38929 


9.239 


24 


31 44 16.52 


3.6330224 


1.40535 


9.130 


26 


34 5 44.97 


3.6224621 


1.41714 


8.994 


28 


36 23 44.51 


3.6113863 


1.42520 


8.814 


30 


38 38 11.23 


+3.5998496 


—1.43003 


—8.538 


32 


40 49 2.74 


3.5879044 


1.43201 


—7.847 


34 


42 56 18.02 


3.5756011 


1.43149 


+8.237 


36 


44 59 57.33 


3.5629877 


1.42877 


8.585 


38 


47 2.00 


3.5501091 


1.42410 


8.753 


40 


48 56 34.33 


+3.5370077 


—1.41772 


+8.857 


42 


50 49 37.39 


3.5237227 


1.40983 


8.928 


44 


52 39 14.95 


3.5102905 


1.40060 


8.978 


46 


54 25 31.32 


3.4967444 


1.39U20 


9.013 


48 


56 8 31.24 


3.4831149 


1.37878 


9.038 


50 


57 48 19.82 


+3.4694297 


—1.36645 


+9.056 


52 


59 25 2.41 


3.4557140 


1.35333 


9.067 


54 


60 58 44.53 


3.4419903 


1.33952 


9.073 


56 


62 29 31.82 


3.4282790 


1.32512 


9.076 


58 


63 57 29.99 


3.4145981 


1.31021 


9.075 


60 


65 22 44.74 


+3.4009637 


—1.29486 


+9.071 


64 


68 5 26.60 


3.3738900 


1.26308 


9.056 


68 


70 38 21.86 


3.3471520 


1.23025 


9.035 


72 


73 2 13.17 


3.3208214 


1.19672 


9.008 


76 


75 17 40.91 


3.2949510 


1.16277 


8.978 


80 


77 25 22.94 


+3.2695785 


—1.12863 


+8.945 


84 


79 25 54.44 


3.2447291 


1.09447 


8.910 


88 


81 19 47.97 


3.2204185 


1.06044 


8.874 


92 


83 7 33.52 


3.1966546 


1.02665 


8.837 


96 


84 49 38.62 


3.1734393 


0.99319 


8.798 


100 


86 26 28.52 


+3.1507694 


—0.96012 


+8.760 


104 


87 58 26.32 


3.1286388 


0.92749 


8.721 


108 


89 25 53.18 


3.1070382 


0.89534 


8.682 


112 


90 49 8.43 


3.0859565 


0.86370 


8.643 


116 


92 8 29.76 


3.0653811 


0.83257 


8.605 



30 



TABLE Ila. 



To. 


Vo. 


Log Ai . 


LogAo. 


Log A3. 


116 


9°2 8 29.76 


+3.0653811 


—0.83257 


+8.605 


120 


93 24 13.33 


3.0452984 


0.80199 


8.567 


124 


94 36 33.98 


3.0256943 


0.77194 


8.529 


128 


95 45 45.25 


3.0065544 


0.74244 


8.491 


132 


96 51 59.60 


2.9878638 


0.71347 


8.454 


136 


97 55 28.43 


+2.9696079 


—0.68505 


+8.418 


140 


98 56 22.24 


2.9517723 


0.65716 


8.382 


144 


99 54 50.68 


2.9343427 


0.62979 


8.346 


148 


100 51 2.62 


2.9173052 


0.60293 


8.311 


152 


101 45 6.25 


2.9006462 


0.57658 


8.276 


156 


102 37 9.12 


+2.8843526 


—0.55071 


+8.242 


160 


103 27 18.23 


2.8684116 


0.52534 


8.209 


164 


104 15 40.03 


2.8528110 


0.50043 


8.176 


168 


J 05 2 20.49 


2.8375388 


0.47598 


8.143 


172 


105 47 25.18 


2.8225838 


0.45198 


8.111 


176 


106 30 59.23 


+2.8079349 


—0.42841 


+8.080 


180 


107 13 7.45 


2.7935817 


0.40526 


8.049 


184 


107 53 54.28 


2.7795141 


0.38253 


8.018 


188 


108 33 23.87 


2.7657223 


0.36020 


7.988 


192 


109 11 40.10 


2.7521971 


0.33826 


7.959 


196 


109 48 46.58 


+2.7389297 


—0.31670 


+7.9.30 


200 


110 24 46.69 


2.7259114 


0.29551 


7.901 


210 


111 50 16.87 


2.6944032 


0.24407 


7.831 


220 


113 9 55.67 


2.6642838 


0.19472 


7.764 


230 


114 24 20.89 


2.6354467 


0.14732 


7.700 


240 


115 34 4.97 


+2.6077961 


—0.10174 


+7.637 


250 


116 39 35.94 


2.5812455 


0.05786 


7.577 


260 


117 41 18.16 


2.5557170 


0.01556 


7.519 


270 


118 39 32.86 


2.5311401 


9.97476 


7.463 


280 


119 34 38.67 


2.5074507 


9.93535 


7.409 


290 


120 26 51.98 


+2.4845910 


—9.89725 


+7.356 


300 


121 16 27.30 


2.4625078 


9.86U38 


7.305 


310 


122 3 37.49 


2.4411532 


9.82467 


7.256 


320 


122 48 34.01 


2.4204831 


9.79006 


7.208 


330 


123 31 27.11 


2.4004569 


9.75648 


7.161 


340 


124 12 25.97 


+2.3810379 


-9.7-2387 


+7.116 


350 


124 51 38.87 


2.3621918 


9.69219 


7.072 


360 


125 29 13.25 


2.3438873 


9.66139 


7.029 


370 


126 5 15.87 


2.3260956 


9.63142 


6.987 


380 


126 39 52.85 


2.3087898 


9.60224 


6.947 


390 


127 13 9.75 


+2.2919450 


—9.57381 


+6.907 


400 


127 45 11.66 


2.2755384 


9.54610 


6.868 


420 


128 45 48.63 


2.2439555 


9.49269 


6.794 


440 


129 42 16.43 


2.2138871 


9.44176 


6.723 


460 


130 35 2.66 


2.1851991 


9.39310 


6.655 



TABLE Ila. 



31 



1 

j to. 


Vo. 


LogAi. 


LogA2. 


LogA^. 


460 


130 3o 2.66 


+2.1851991 


—9.39310 


+96.655 


480 


131 24 30.82 


2.1577741 


9.34654 


6.589 


500 


132 11 1.09 


2.1315086 


9.30188 


6.527 


1 520 


132 54 50.84 


2.1063114 


9.25901 


6.467 


540 


133 36 15.19 


2.0821011 


9.21777 


6.409 


560 


134 15 27.33 


+2.0588051 


—9.17805 


+96.353 


580 


134 52 38.80 


2.0363588 


9.13976 


6.299 


600 


135 27 59.81 


2.0147037 


9.10278 


6.247 


640 


136 33 45.52 


1.9735615 


9.03246 


6.148 


680 


137 33 45.39 


1.9350140 


8.96649 


6.055 


720 


138 28 48.27 


+1.8987593 


—8.90438 


+95.968 


760 


139 19 33.81 


1.8645446 


8.84571 


5.885 


800 


140 6 34.57 


1.8321564 


8.79012 


5.807 


850 


141 45.22 


1.7939648 


8.72451 


5.714 


900 


141 50 30.05 


1.7580440 


8.66275 


5.627 


950 


142 36 24.37 


+1.7241428 


—8.60441 


+95.544 


1000 


143 18 57.20 


1.6920492 


8.54915 


5.466 


1050 


143 58 32.66 


1.6615826 


8.49.665 


5.392 


1100 


144 35 30.95 


1.6325881 


8.44666 


5.321 


1150 


145 10 9.20 


1.6049315 


8.39896 


5.254 


1200 


145 42 41.98 


+1.5784963 


—8.35333 


+95.189 


1250 


146 13 21.82 


1.5531804 


8.30962 


5.127 


1300 


146 42 19.55 


1.5288937 


8.26767 


5.068 


1350 


147 9 44.57 


1.5055568 


8.22735 


5.011 


1400 


147 35 45.11 


1.4830989 


8.18853 


4.956 


1450 


148 28.40 


+1.4614567 


—8.15110 


+94.903 


1500 


148 24 0.83 


1.4405738 


8.11498 


4.851 


1600 


149 7 55.10 


1.4008865 


8.04631 


4.754 


1700 


149 48 6.25 


1.3636849 


7.98190 


4.663 


1800 


150 25 5.10 


1.3286785 


7.92126 


4.576 


1900 


150 59 16.75 


+1.2956243 


, —7.86398 


+94.495 


2000 


151 31 1.89 


1.2643177 


7.80971 


4.418 


2100 


152 37.76 


1.2345845 


7.75814 


4.345 


2200 


152 28 18.85 


1.2062750 


7.70903 


4.275 


2300 


152 54 17.45 


1.1792601 


7.66216 


4.208 


2400 


153 18 44.05 


+1.1534272 


—7.61732 


+94.145 


2500 


153 41 47.70 


1.1286779 


7.57435 


4.084 


2600 


154 3 36.21 


1.1049254 


7.53310 


4.025 


2700 


154 24 16.39 


1.0820930 


7.49344 


3.969 


2800 


154 43 54.21 


1.0601125 


7.45526 


3.914 


2900 


155 2 34.93 


+1.0389230 


—7.41844 


+93.862 


3000 


155 20 23.19 


1.0184698 


7.38289 


3.811 


3200 


155 53 38.39 


0.9795803 


7.31529 


3.715 


3400 


156 24 7.80 


0.9431040 


7.25186 


3.625 


3600 


156 52 14.00 


0.9087603 


7.19213 


3.540 



32 



TABLE Ila. 



To. 


fo- 


Log Ai . 


LogA2. 


Log As- 


3600 


15°6 52 14.00 


4-0.9087603 


—97.19213 


+93.540 


3800 


157 18 15.42 


0.8763145 


7.13568 


3.459 


4000 


157 42 27.29 


0.8455688 


7.08218 


3.383 


4200 


158 5 2.33 


0.8163545 


7.03133 


3.311 


4400 


158 26 11.25 


0.7885269 


6.98289 


3.242 


4600 


158 46 3.15 


-^0.7619607 


—96.93664 


+93.176 


4800 


159 4 45.83 


0.7365469 


6.89238 


3.113 


5000 


159 22 25.99 


0.7121902 


6.84996 


3.053 


5200 


159 39 9.45 


0.6888063 


6.80923 


2.995 


5600 


160 10 6.00 


0.6446674 


6.73234 


2.885 


6000 


160 38 9.17 


+0.6036264 


—96.66082 


+92.783 


6400 


161 3 45.36 


0.5652780 


6.59398 


2.688 


6800 


161 27 15.57 


0.5292915 


6.53125 


2.599 


7200 


J 61 48 56.78 


0.4953934 


6.47215 


2.514 


7600 


162 9 2.89 


0.4633554 


6.41629 


2.435 


8000 


162 27 45.39 


+0.4329843 


—96.36332 


f92.359 


8400 


162 45 13.90 


0.4041157 


6.31297 


2.287 


8800 


163 1 36.52 


0.3766081 


6.26499 


2.219 


9200 


163 17 0.16 


0.3503393 


6.21916 


2.154 


9600 


163 31 30.72 


0.3252029 


6.17531 


2.091 


10000 


163 45 13.32 


+0.3011054 


—96.13326 


+92.031 


10500 


164 1 20.80 


0.2723199 


6.08303 


1.959 


11000 


164 16 27.66 


0.2448894 


6.03516 


1.891 


11500 


164 30 40.23 


0.2186921 


5.98944 


1.826 


12000 


164 44 3.94 


0.1936223 


5.94568 


1.764 


13000 


165 8 42.90 


+0.1465042 


—95.86343 


+91.646 


14000 


165 30 55.26 


0.1029147 


5.78733 


1.538 


15000 


165 51 4.63 


0.0623627 


5.71652 


1.437 


16000 


166 9 29.58 


0.0244528 


5.65032 


1.342 


17000 


166 26 24.88 


9.9888624 


5.58817 


1.254 


18000 


166 42 2.53 


+9.9553241 


—95.52959 


+91.170 


19200 


166 59 18.90 


9.9174751 


5.46348 


1.076 


20400 


167 15 11.32 


9.8819393 


5.40141 


90.987 


21600 


167 29 51.00 


9.8484507 


5.34290 


90.904 


22800 


167 43 27.11 


9.8167866 


5.28758 


90.825 


24000 


167 56 7.28 


+9.7867585 


—95.23512 


+90.750 


26000 


168 15 26.77 


9.7399215 


5.15328 


90.633 


28000 


168 32 51.95 


9.6965794 


5.07755 


90.525 


30000 


168 48 41.17 


9.6562474 


5.00706 


90.424 


32000 


169 3 8.84 


9.6185347 


4.94116 


90.330 


34000 


169 16 26.46 


+9.5831221 


—94.87926 


+90.242 


36000 


169 28 43.36 


9.5497452 


4.82093 


90.159 


38000 


169 40 7.19 


9.5181828 


4.76576 


90.080 


40000 


169 50 44.28 


9.4882481 


4.71343 


90.005 



TABLE Ilia. 



33 



■n 


Log ^. 


Log Diff. 


V 


Log 11. 


Log Diff. 


V 


Log fi. 


Log Diff. 


0.00 
.01 
.02 


0.00000 00 
.00000 18 
.00000 72 


1.556 
1.857 


0.30 
.31 
.32 


0.00167 33 
.00179 01 
.00191 12 


3.0594 
.0754 
.0910 


0.60 
.61 
.62 


0.00735 26 
.00763 61 
.00792 74 


3.4468 
.4585 
.4703 


0.03 
.04 
.05 


0.00001 62 
.00002 89 
.00004 52 


2.0354 
.1614 
.2589 


0.33 
.34 
.35 


0.00203 67 
.00216 66 
.00230 10 


3 1062 
.1211 
.1356 


0.63 
.64 
.65 


0.00822 68 
.00853 45 
.00885 08 


3.4822 
.4941 
.5061 


0.06 

.07 
.08 


.00006 52 
.00008 88 
.00011 61 


2.3385 
.4057 
.4639 


0.36 
.37 
.38 


0.00243 99 
.00258 34 
.00273 15 


3.1498 
.1638 
.1774 


0.66 
.67 
.68 


0.00917 59 
.00951 03 
.00985 42 


3.5182 
.5304 
5427 


1 0.09 

i .10 

.11 


0.00014 70 
.00018 16 
.00021 99 


2.5152 
.5617 
.6031 


0.39 
.40 
.41 


0.00288 43 
.00304 20 
.00320 45 


3.1911 
.2044 
.2175 


0.69 
.70 
.71 


0.01020 81 
.01057 23 
.01094 73 


3.5551 
.5677 
.5805 


0.12 
.13 
.14 


0.00026 18 
.00030 74 
.00035 68 


2.6410 
.6767 
.7097 


0.42 
.43 
.44 


0.00337 20 
.00354 45 
.00372 22 


3.2304 
.2433 
.2557 


0.72 
.73 
.74 


0.01133 35 
.01173 15 
.01214 19 


3.5934 
.6066 
.6200 


0.15 
.16 
.17 


0.00040 99 
.00046 68 
.00052 75 


2.7404 
.7694 
.7966 


0.45 
.46 
.47 


0.00390 50 
.00409 31 
.00428 67 


3.2681 
.2807 
.2930 


0.75 
.76 

.77 


0.01256 52 
.01300 22 
.01345 36 


3.6336 
.6476 
.6618 


0.18 
.19 
.20 


0.00059 20 
.00066 03 
.00073 25 


2.8222 
.8466 
.8701 


0.48 
.49 
.50 


0.00448 58 
.00469 06 
.00490 11 


3.3053 
•3173 
.3293 


0.78 
.79 
.80 


0.01392 02 
.01440 31 
.01490 32 


3.6765 
.6915 
.7070 


0.21 
.22 

.23 


0.00080 86 
.00088 86 
.00097 25 


2.8924 
.9135 
.9340 


0.51 
.52 
.53 


0.00511 75 
.00533 98 
.00556 83 


3.3411 

.3529 
.3647 


0.81 
.82 
.83 


0.01542 18 
.01596 03 
.01652 02 


3.7231 
.7397 
.7570 


0.24 

.25 
.26 


0.00106 04 
.00115 23 
.00124 83 


2.9538 
.9729 
.9914 


0.54 
.55 
.56 


0.00580 30 
.00604 41 
.00629 19 


3.3764 

.3882 
.4000 


0.84 

.85 
.86 


0.01710 33 
.01771 19 
.01834 86 


3.7751 
.7942 
.8144 


0.27 

.28 
.29 


.00134 84 
.00145 25 
.00156 08 


3.0090 
.0261 
.0430 


0.57 
.58 
.59 


0.00654 65 
.00680 80 
.00707 66 


3.4117 
.4233 
.4350 


0.87 
.88 
.89 


0.01901 65 
.01971 95 
.02046 29 


3.8360 
.8593 
.8846 


0.30 
.31 
.32 


0.00167 33 
.00179 01 
.00191 12 


3.0594 
.0754 
.0910 


0.60 
.61 


0.00735 26 
.00763 61 
.00792 74 


3.4468 
.4585 
.4703 


0.90 
.91 
.92 


0.02125 29 
.02209 92 
.02301 60 


3.9128 
.9452 

i 



34 



TABLE IV, 





m sin z* = sin (z — y) 


m and 


q positive 








'i 


's 


0' 




^ 


^m 




z"" 


q 


^' 


o 


m" 


m' 


m' 


vf 


m" 


m! 


m 


' 


«/' 


o 






O 4 


o 


o 


O 1 


o 


o / 


^ 




^ 


1 


4.2976 


9.9999 


1 


1 20 


1 20 


89 40 


89 40 


177 37 


180 


5o 


181 b 


2 


3.3950 


9.9996 


2 


2 40 


2 40 


89 20 


89 20 


175 14 


181 


51 


182 


3 


2.8675 


9.9992 


3 


4 


4 


89 


89 


172 52 


182 


46 


183 


4 


2.4938 


9.9986 


4 


5 20 


5 20 


88 40 


88 40 


170 28 


183 


42 


184 


5 


2.2044 


9.9978 


5 


6 41 


6 41 


88 19 


88 19 


168 5 


184 37 


185 


6 


1.9686 


9.9968 


6 


8 1 


8 1 


87 59 


87 59 


165 41 


185 


32 


186 


7 


1.7698 


9.9957 


7 1 


9 22 


9 22 


87 38 


87 38 


163 18 


186 


28 


186 59 


8 


1.5981 


9.9943 


8 1 


10 42 


10 42 


87 18 


87 18 


160 52 


187 


23 


187 59 


9 


1.4473 


9.9928 


9 2 


12 3 


12 3 


86 57 


86 57 


158 28 


188 


18 


188 58 


10 


1.3130 


9.9911 


10 3 


13 25 


13 25 


86 35 


86 35 


156 3 


189 


13 


189 57 


11 


1.1922 


9.9892 


11 5 


14 46 


14 46 


86 14 


86 14 


153 37 


190 


9 


190 56 


12 


1.0824 


9.9871 


12 7 


16 8 


16 8 


85 b-l 


85 52 


151 10 


191 


4 


191 54 


13 


0.9821 


9.9848 


13 9 


17 31 


17 31 


85 29 


85 29 


148 43 


191 


59 


192 52 


14 


0.8898 


9.9823 


14 12 


18 53 


18 53 


85 7 


85 7 


146 14 


192 


54 


193 49 


15 


0.8045 


9.9796 


15 16 


20 17 


20 17 


84 43 


84 43 


143 45 


193 


49 


194 46 


16 


0.7254 


9.9767 


16 20 


21 40 


21 40 


84 20 


84 20 


141 14 


194 


44. 


195 42 


17 


0.6518 


9.9736 


17 26 


23 5 


23 5 


83 55 


83 55 


138 42 


195 


39 


196 38 


18 


0.5830 


9.9702 


18 33 


24 30 


24 30 


83 30 


83 30 


136 9 


196 


33 


197 33 


19 


0.5185 


9.9667 


19 41 


25 56 


25 56 


83 4 


83 4 


133 34 


197 


28 


198 28 


20 


0.4581 


9.9629 


20 51 


27 23 


27 23 


82 37 


82 37 


130 58 


198 


23 


199 22 


21 


0.4013 


9.9588 


22 2 


28 50 


28 50 


82 10 


82 10 


128 19 


199 


17 


200 15 


22 


0.3479 


9.9545 


23 15 


30 19 


30 19 


81 41 


81 41 


125 38 


200 


11 


201 8 


23 


0.2976 


9.9499 


24 31 


31 49 


31 49 


81 11 


81 11 


122 55 


201 


6 


202 


24 


0.2501 


9.9451 


25 49 


33 20 


33 20 


80 40 


80 40 


120 9 


202 





202 51 


25 


0.2053 


9.9400 


27 10 


34 53 


34 53 


80 7 


80 7 


117 20 


202 


54 


203 42 


26 


0.1631 


9.9345 


28 35 


36 28 


36 28 


79 32 


79 32 


114 27 


203 


47 


204 32 


27 


0.1232 


9.9287 


30 4 


38 5 


38 5 


78 55 


78 55 


111 30 


204 


41 


205 22 


28 


0.0857 


9.9226 


31 38 


39 45 


39 45 


78 15 


78 15 


108 27 


205 


35 


206 11 


29 


0.0503 


9.9161 


33 18 


41 27 


41 27 


77 33 


77 33 


105 19 


206 


28 


207 


30 


0.0170 


9.9092 


35 5 


43 13 


43 13 


76 47 


76 47 


102 3 


207 


21 


207 48 


31 


9.9857 


9.9019 


37 1 


45 4 


45 4 


75 56 


75 56 


98 37 


208 


14 


208 36 


32 


9.9565 


9.8940 


39 9 


47 1 


47 1 


74 59 


74 59 


95 


209 


6 


209 24 


33 


9.0292 


9.8856 


41 33 


49 6 


49 6 


73 54 


73 54 


91 6 


209 


58 


210 11 


1 34 


9.9040 


9.8765 


44 21 


51 22 


51 22 


72 38 


72 38 


86 49 


210 


50 


210 58 


1 35 


9.8808 


9.8665 


47 47 


53 58 


53 58 


71 2 


71 2 


81 53 


211 


41 


211 46 


36 


9.8600 


9.8555 


52 31 


57 13 


57 13 


68 47 


68 47 


75 40 


212 


32 


212 33 


9' 


9.8443 


9.8443 


63 26 


63 26 


63 26 


63 26 


63 26 


63 26 


213 


15 


213 15 


q' = 


36° 52' 11.64" 






sin c 


f = OS. 







TABLE IV 



35 









in sin z* = 


=:sin(a: + 9). 


m 


and q positive. 










'g 


'^ 






2' 






0° 






^m 






2 


IT 


9. 


-i 


fcc 


m 


' 


m 


/ 


m 


' 


m 




m 


/ 


m 


' 


m 




m 


^ 






^ 




o 









o 




o 




^ 




^ 




^ 


1 


4.2976 


9.9999 


2 


23 


90 


20 


90 


20 


178 


40 


178 


40 


179 





359 





359 


2 


3.3950 


9.9996 


4 


46 


90 


40 


90 


40 


177 


20 


177 


20 


178 





358 





358 9 


3 


2.8675 


9.9992 


7 


8 


91 





91 





175 





175 





177 





357 





357 14 


4 


2.4938 


9.9986 


9 


32 


91 


20 


91 


20 


174 


40 


174 


40 


176 





356 





356 18 


5 


2.2044 


9.9978 


11 


55 


91 


41 


91 


41 


173 


19 


173 


19 


175 





355 





355 23 


6 


1.9686 


9.9968 


14 


19 


92 


1 


92 


1 


171 


59 


171 


59 


174 





354 





354 28 


7 


1.7698 


9.9957 


16 


42 


92 


22 


92 


22 


170 


38 


170 


38 


172 


59 


353 


1 


353 32 


8 


1.5981 


9.9943 


19 


7 


92 


42 


92 


42 


169 


18 


169 


18 


171 


59 


352 


1 


352 37 


9 


1.4473 


9.9928 


21 


32 


93 


3 


93 


3 


167 


57 


167 


57 


170 


58 


351 


2 


351 42 


10 


1.3130 


9.9911 


23 


57 


93 


25 


93 


25 


166 


35 


166 


35 


169 


57 


350 


3 


350 47 


11 


1.1922 


9.9892 


26 


23 


93 


46 


93 


46 


165 


14 


165 


14 


168 


55 


349 


4 


349 51 


12 


1.0824 


9.9871 


28 


50 


94 


8 


94 


8 


163 


52 


163 


52 


167 


54 


348 


6 


348 56 


13 


0.9821 


9.9848 


31 


17 


94 


31 


94 31 


162 


29 


162 


29 


166 


51 


347 


,8 


348 1 


14 


0.8898 


9.9823 


33 


46 


94 


53 


94 


53 


161 


7 


161 


7 


165 


48 


346 


11 


347 6 


15 


0.8045 


9.9796 


36 


15 


95 


17 


95 


17 


159 


43 


159 


43 


164 


44 


345 


14 


346 11 


16 


0.7254 


9.9767 


38 


46 


95 


40 


95 


40 


158 


20 


158 


20 


163 


40 


344 


18 


345 16 


17 


0.6518 


9.9736 


41 


18 


96 


5 


96 


5 


156 


55 


156 


55 


162 


34 


343 


22 


344 21 


18 


0.5830 


9.9702 


43 


51 


96 


30 


96 


30 


155 


30 


155 


30 


161 


27 


342 


27 


343 27 


19 


0.5185 


9.9667 


46 


26 


96 


56 


96 


56 


154 


4 


154 


4 


160 


19 


341 


32 


342 32 


20 


0.4581 


9.9629 


49 


2 


97 


23 


97 


23 


1.52 


37 


152 


37 


159 


9 


340 


38 


341 37 


21 


0.4013 


9.9588 


51 


41 


97 


50 


97 


50 


151 


10 


151 


10 


157 


58 


339 


45 


340 43 


22 


0.3479 


9.9545 


54 22 


98 


19 


98 


19 


149 


41 


149 


41 


156 


45 


338 


52 


339 49 


23 


0.2976 


9.9499 


57 


5 


98 


49 


98 


49 


148 


11 


148 


11 


155 


29 


338 





338 54 


24 


0.2501 


9.9451 


59 


51 


99 


20 


99 


20 


146 


40 


146 


40 


154 


11 


337 


9 


338 


25 


0.2053 


9.9400 


62 


40 


99 


53 


99 


53 


145 


7 


145 


7 


152 


50 


336 


18 


337 6 


26 


0.1631 


9.9345 


65 


33 


100 


28 


100 


28 


143 


32 


143 


32 


151 


25 


335 


28 


336 13 


27 


0.1232 


9.9287 


68 


30 


101 


5 


101 


5 


141 


55 


141 


55 


149 


56 


334 


38 


335 19 


28 


0.0857 


9.9226 


71 


33 


101 


45 


101 


45 


140 


15 


140 


15 


148 


22 


333 


49 


334 25 


29 


0.0503 


9.9161 


74 


41 


102 


27 


102 


27 


138 


33 


138 


33 


146 


42 


333 





333 32 


30 


0.0170 


9.9092 


77 


57 


103 


13 


103 


13 


136 


46 


136 


46 


144 55 


332 


12 


332 39 


31 


9.9857 


9.9019 


81 


23 


104 


4 


104 


4 


134 56 


134 


56 


142 


59 


331 


24 


331 46 


32 


9.9565 


9.8940 


85 





105 


1 


105 


1 


132 


59 


132 


59 


140 


51 


330 


36 


330 54 


33 


9.9292 


9.8856 


88 


54 


106 


6 


106 


6 


130 


54 


130 


54 


138 


27 


329 


49 


330 2 


34 


9.9040 


9.8765 


93 


11 


107 


22 


107 


22 


128 


38 


128 


38 


135 


38 


329 


2 


329 10 


35 


9.8808 


9.8665 


98 


7 


108 


58 


108 


58 


126 


2 


126 


2 


132 


13 


328 


14 


328 19 


36 


9.8600 


9.8555 


104 


20 


111 
116 


13 
34 


111 


13 


122 


47 


122 


47 


127 


29 


327 


27 


327 28 


?' 


9.8443 


9.8443 


116 34 


116 


34 


116 34 


116 34 


116 


H 


326 


45 


326 45 




?' = 


:36° 


52' 


LI. 64" 












sin q' = 0.6 



































36 



TABLE Va. 



- ■ 


A. 


Diff. 


B. 


Diff. 


B'. 


Diff. 1^ 





- '^.00 


—9.60 


— o'.ooo 


—11 


— o'.ooo 


—34 


1 


9.00 


9.00 


0.011 


11 


0.034 


34 


2 


17.99 


8.98 


0.023 


12 


0.067 


33 


3 


26.95 


8.95 


0.034 


11 


0.101 


34 


4 


35.88 


8.91 


0.045 


11 


0.134 


33 


5 


— 44.77 


—8.87 


—0.057 


—12 


—0.167 


—33 


6 


53.61 


8.80 


0.068 


11 


0.200 


33 


7 


62.37 


8.73 


0.080 


12 


0.232 


32 


8 


71.07 


8.65 


0.092 


12 


0.263 


31 


9 


79.67 


8.56 


0.104 


12 


0.294 


31 


10 


— 88.18 


—8.46 


—0.117 


—13 


—0.324 


—30 


11 


96.58 


8.34 


0.129 


12 


0.353 


29 


12 


104.86 


8.22 


0.142 


13 


0.382 


29 


13 


113.01 


8.08 


0.156 


14 


0.409 


27 


14 


121.02 


7.94 


0.169 


13 


0.436 


27 


15 


—128.88 


—7.79 


—0.183 


—14 


—0.461 


—25 


16 


136.59 


7^62 


0.197 


14 


0.486 


25 


17 


144.12 


7.43 


0.211 


14 


0.509 


23 


18 


151.47 


7.27 


0.226 


15 


0.531 


22 


19 


158.63 


7.08 


0.241 


15 


0.552 


21 


20 


—165.60 


—6.86 


—0.256 


—15 


—0.571 


—19 


21 


172.35 


6.65 


0.271 


15 


0.590 


19 


22 


178.89 


6.43 


0.287 


16 


0.606 


16 


23 


185.20 


6.20 


0.303 


16 


0.622 


16 


24 


191.28 


5.96 


0.319 


16 


0.636 


14 


25 


—J 97.11 


—5.71 


—0.336 


—17 


—0.648 


—12 


26 


202.69 


5.45 


0.352 


16 


0.659 


10 


27 


208.00 


5.18 


0.369 


17 


0.668 


9 


28 


213.05 


4.91 


0.386 


17 


0.676 


7 


29 


217.81 


4.63 


0.403 


17 


0.682 


6 


30 


—222.30 


—4.34 


—0.419 


—16 


—0.687 


— 4 


31 


226.48 


4.04 


0.436 


17 


0.690 


3 


32 


230.37 


3.74 


0.453 


17 


0.692 


1 


33 


233.95 


3.42 


0.470 


17 


0.692 





34 


237.21 


3.10 


0.486 


16 


0.691 


4-2 


35 


—240.15 


—2.78 


—0.502 


—16 


—0.688 


+ 4 


36 


242.76 


2.45 


0.518 


16 


0.683 


5 


37 


245.04 


2.11 


0.534 


16 


0.677 


6 


38 


246.98 


1.77 


0.549 


15 


0.670 


8 


39 


248.57 


1.41 


0.564 


15 


0.661 


9 


40 


—249.80 


—1.06 


—0.578 


—14 


—0.651 


+11 


41 


250.68 


0.70 


0.591 


13 


0.639 


12 


42 


251.20 


0.33 


0.604 


12 


0.627 


13 



TABLE Va 



37 



X. 


A. 


Diff. 


B. 


Diff. 


B'. 


Diff. 


42 


—251.20 


— 0.33 


— 6'.604 


— 12 


— 6'.627 


+13 


43 


251.34 


+ 0.04 


0.615 


11 


0.613 


15 


44 


251.11 


^ 0.42 


0.626 


11 


0.597 


16 


45 


250.50 


0.80 


0.636 


10 


0.580 


17 


46 


249.51 


1.18 


0.645 


8 


0.563 


18 


47 


—248.13 


+ 1.57 


—0.652 


— 7 


—0.544 


+19 


48 


246.36 


1.96- 


0.659 


6 


0.524 


20 


49 


244.20 


2.36 


0.664 


4 


0.503 


21 


50 


241.64 


2.76 


0.667 


3 


0.482 


22 


51 


238.68 


3.16 


0.669 


1 


0.459 


23 


52 


—235.31 


+ 3.57 


—0.669 


+ 1 


—0.436 


+23 


53 


231.54 


3.98 


0.667 


2 


0.412 


24 


54 


227.35 


4.39 


0.664 


4 


0.387 


25 


55 


222.76 


4.80 


0.659 


6 


0.361 


26 


56 


217.75 


5.22 


0.651 


9 


0.335 


26 


57 


—212.32 


+ 5.64 


—0.641 


+ 11 


—0.309 


+26 


58 


206.47 


6.06 


0.629 


13 


0.282 


27 


59 


200.20 


6.47 


0.615 


15 


0.255 


27 


60 


193.52 


6.90 


0.598 


18 


0.227 


28 


61 


186.40 


7.32 


0.579 


20 


0.200 


27 


62 


—178.87 


+ 7.74 


-0.557 


+ 23 


—0.172 


+28 


63 


170.91 


8.17 


0.532 


26 


0.144 


28 


64 


162.52 


8.60 


0.504 


29 


0.116 


28 


65 


153.70 


9.03 


0.474 


32 


0.088 


28 


66 


144.46 


9.45 


0.440 


35 


0.061 


27 


67 


—134.79 


+ 9.88 


—0.403 


+ 38 


—0.033 


+28 


68 


124.69 


10.31 


0.363 


41 


—0.006 


27 


69 


114.16 


10.74 


0.320 


45 


+0.021 


27 


70 


103.20 


11.17 


0.273 


49 


0.048 


27 


71 


91.81 


11.60 


0.222 


52 


0.074 


26 


72 


— 80.00 


+12.03 


—0.168 


+ 56 


+0.099 


+25 


73 


67.75 


12.46 


0.110 


59 


0.124 


25 


74 


55.07 


12.89 


0.049 


63 


0.148 


24 


75 


41.97 


13.32 


+0.016 


67 


0.172 


24 


76 


28.43 


13.72 


0.086 


71 


0.195 


22 


77 


— 14.47 


+14.18 


+0.159 


+ 75 


+0.216 


+21 


78 


0.07 


14.61 


0.237 


80 


0.237 


21 


79 


+ 14.76 


15.04 


0.319 


84 


0.257 


20 


80 


30.02 


15.47 


0.405 


88 


0.276 


19 


81 


45.70 


15.89 


0.496 


93 


0.294 


18 


82 


+ 61.80 


f 16.32 


+0.591 


+ 97 


+0.311 


+16 


83 


78.34 


16.76 


0.691 


102 


0.326 


15 


84 


95.32 


17.19 


0.795 


106 


0.340 


13 ' 



38 



TABLE \c 



X. 


A. 


Diff. 


B. 


Diff. 


B'. 


Diff. 


8°4 


+ 90.32 


+17.19 


+ 0795 


+106 


+6'.340 


+ 13 


85 


112.72 


17.62 


0.904 


111 


0.352 


12 


86 


130.56 


18.06 


1.018 


116 


0.363 


10 


87 


148.84 


18.49 


1.137 


121 


0.373 


9 


88 


167.54 


18.92 


1.261 


126 


0.381 


7 


89 


+ 186.69 


+19.36 


+ 1.390 


+132 


+0.386 


+ 5 


90 


206.27 


19.80 


1.525 


137 


0.390 




91 


226.29 


20.24 


1.665 


142 


0.392 


1 


92 


246.75 


20.68 


1.810 


148 


0.392 


— 1 


93 


267.65 


21.13 


1.961 


154 


0.390 


3 


94 


+ 289.01 


+21.58 


+ 2.118 


+159 


+0.385 


— 6 


95 


310.82 


22.03 


2.280 


165 


0.378 


8 


96 


333.08 


22.49 


2.449 


171 


0.368 


11 


97 


355.80 


22.95 


2.623 


178 


0.355 


14 


98 


378.99 


23.42 


2.805 


184 


0.339 


17 


99 


+ 402.65 


+23.89 


+ 2.992 


+191 


+0.320 


— 21 


100 


426.78 


24.37 


3.187 


198 


0.297 


25 


101 


451.40 


24.86 


3.388 


204 


0.270 


28 


102 


476.51 


25.36 


3.596 


212 


0.240 


32 


103 


502.12 


25.86 


3.812 


220 


0.205 


37 


104 


+ 528.24 


+26.38 


+ 4.036 


+227 


+0.165 


— 42 


105 


554.88 


26.90 


4.267 


235 


0.121 


47 


106 


582.04 


27.43 


4.506 


240 


0.071 


53 


107 


609.75 


27.99 


4.755 


250 


+0.015 


59 


108 


638.02 


28.55 


5.012 


261 


—0.048 


66 


109 


+ 666.85 


+29.11 


+ 5.278 


+271 


—0.117 


— 72 


110 


696.27 


29.72 


5.554 


281 


0.193 


80 


111 


726.29 


30.33 


5.841 


292 


0.278 


89 


112 


756.93 


30.96 


6.138 


302 


0.371 


98 


113 


788.21 


31.61 


6.446 


314 


0.474 


108 


114 


+ 820.15 


+32.28 


+ 6.766 


+326 


—0.587 


—119 


115 


852.77 


32.98 


7.099 


339 


0.712 


131 


116 


886.11 


33.70 


7.445 


353 


0.849 


144 


117 


920.18 


34.45 


8.806 


368 


1.000 


158 


118 


955.02 


35.22 


8.181 


383 


1.166 


174 


119 


+ 990.65 


+36.05 


+ 8.572 


+399 


—1.348 


—191 


120 


J 027.13 


36.91 


8.980 


417 


1.548 


209 


121 


1064.47 


37.79 


9.407 


436 


1.767 


230 


122 


1102.71 


38.73 


9.853 


456 


2.009 


253 


123 


1141.93 


39.71 


10.320 


478 


2.274 


278 


124 


+1182.14 


+40.74 


+10.809 


+501 


—2.566 


—306 


125 


1223.41 


41.82 


11.323 


527 


2.886 


336 


126 


1265.78 


42.96 


11.863 


554 


3.239 


370 i 



TABLE Va. 



39 



X. 


A. 


Diflf. 


B. 


Diff. 


B'. 


Diff. 


126 


+1260.78 


+ 42.96 


+ ll863 


+ 0.554 


— 3^239 


— 0.370 


127 


1309.33 


44.16 


12.431 


0.584 


3.627 


0.408 


128 


1354.11 


45.43 


13.031 


0.616 


4.055 


0.449 


129 


1400.20 


46.78 


13.663 


0.651 


4.526 


0.496 


130 


1447.67 


48.20 


14.333 


0.690 


5.047 


0.547 


131 


+1496.61 


+ 49.72 


+ 15.043 


+ 0.731 


— 5.621 


— 0.605 


132 


1547.11 


51.33 


15.796 


0.777 


6.257 


0.669 


133 


1599.28 


53.04 


16.597 


0.827 


6.960 


0.741 


134 


1653.20 


54.87 


17.451 


0.883 


7.739 


0.821 


135 


1709.02 


56.82 


18.363 


0.945 


8.603 


0.912 


136 


+1766.84 


+ 58.91 


+ 19.341 


+ 1.013 


— 9.563 


— 1.014 


137 


1826.84 


61.15 


20.389 


1.088 


10.631 


1.128 


138 


1889.15 


63.55 


21.517 


1.171 


11.820 


1.258 


139 


1953.95 


66.14 


22.732 


1.265 


13.148 


1.406 


140 


2021.43 


68.92 


24.047 


1.371 


14.633 


1.573 


141 


+2091.79 


+ 71.90 


+ 25.475 


+ 1.490 


— 16.295 


— 1.765 


142 


2165.28 


75.15 


27.027 


1.623 


18.163 


1.984 


143 


2242.15 


78.65 


28.722 


1.774 


20.263 


2.234 


144 


2322.68 


82.47 


30.576 


1.946 


22.631 


2.523 


145 


2407.20 


86.58 


32.615 


2.143 


25.309 


2.856 


146 


+2496.06 


+ 91.16 


+ 34.862 


+ 2.368 


— 28.344 


— 3.242 


147 


2589.66 


96.11 


37.351 


2.626 


31.794 


3.713 


148 


2688.45 


101.56 


40.115 


2.924 


35.730 


4.224 


149 


2792.96 


107.54 


43.199 


3.272 


40.233 


4.836 


150 


2903.74 


114.13 


46.659 


3.677 


45.403 


5.566 


151 


+3021.46 


+121.43 


+ 50.553 


+ 4.153 


— 51.366 


— 6.437 


152 


3146.88 


129.53 


54.966 


4.717 


58.267 


7.469 


153 


3280.84 


138.56 


59.987 


5.385 


66.295 


8.705 


154 


3424.37 


148.67 


' 65.737 


6.185 


75.677 


10.202 


155 


3578.59 


160.01 


72.357 


7.155 


86.700 


12.024 


156 


+3744.88 


+172.81 


+ 80.042 


+ 8.328 


— 99.726 


— 14.260 


157 


3924.79 


187.33 


89.014 


9.767 


115.221 


17.023 


158 


4120.22 


203.89 


99.577 


11.548 


133.773 


20.471 


159 


4333.38 


222.87 


112.111 


13.777 


156.174 


24.815 


160 


4566.94 


244.78 


127.132 


16.603 


183.404 


30.348 


161 


+4824.14 


+244.78 


+145.317 


+20.209 


—216.860 


— 37.483 


162 


5108.93 


270.26 


167.550 


24.869 


258.371 


46.802 


163 


5426.19 


300.11 


195.056 


31.062 


310.464 


59.156 


164 


5782.01 


335.39 


229.674 


39.353 


376.683 


75.318 


165 


6184.14 


377.50 


273.762 


50.636 


462.100 


98.618 


166 


+6642.49 


+428.33 


+330.946 


+66.405 


—574.089 


—130.816 


167 


7170.07 


490.43 


406.573 


88.993 


723.733 


177.025 


168 


7784.18 


567.43 


508.933 


122.256 


928.140 


246.403 


16'J 


8508.45 




651.086 




1214.530 





CONSTANTS. 



Log. 

Attractive force of the Sun, h in terms of radius, .0172021 8.2355814 

k in seconds, 3548".18761 3.5500066 

Length of the Sidereal Year (Hansen and Olufsen), 365^2563582 2.5625978 

Length of the Tropical Year, 1850, 365^2422008 2.5625809 

Horizontal equatorial parallax of the Sun (Encke),* 8".5776 0.9333658 

Constant of Aberration (Struve), 20".4451 1.3105892 
Time required for light to pass from. the Ssin to 

the Earth, 497^827 2.6970785 

Radius of Circle in Seconds of arc, 206264".806 5.3144251 

in Seconds of time, 13750^987 4.1383339 

Sin r 0.000004848137 4.6855749 

Cu-cumference of Circle in Seconds of arc, 1296000" 6.1126050 

in Seconds of time, 86400' 4.9365137 

in terms of diameter, 7T 3.14159265 0.4971499 

General Precession (Struve) 50".2411 + 0''.0002268^ 

Obliquity of the ecliptic (Struve and Peters), 23° 27'54".22 - 0.4645 z!— .0000014^ 

in which t is the number of years after 1800 

DaHy precession, 1850, 0".1375837 9.1385669 

Modulus of Common Logarithms, M 0.4342945 9.6377843 



* The Constants of Parallax, Aberration, etc., are those used in the American 
the authority for them may be found by reference to the volume for 1855. 
(40) 



kemeris, and 



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LIBRARY OF CONGRESS 




iiiiiililliiiiiljilliilii 

003 536 198 4 



